diff --git a/src/Quadmath.jl b/src/Quadmath.jl
index efcbe07..b3af96a 100644
--- a/src/Quadmath.jl
+++ b/src/Quadmath.jl
@@ -226,14 +226,15 @@ for Ti in (UInt8, UInt16, UInt32, UInt64, UInt128)
     # `Float128(typemax(Ti))+1` is exactly representable
     @eval begin
         function trunc(::Type{$Ti},x::Float128)
-            if !signbit(x) && x < Float128(typemax($Ti)) + one(Float128)
+            if (!signbit(x) && x < Float128(typemax($Ti)) + one(Float128)) || (x == 0)
                 return unsafe_trunc($Ti,x)
             else
                 throw(InexactError(:trunc, $Ti, x))
             end
         end
         function (::Type{$Ti})(x::Float128)
-            if (!signbit(x) && x <= Float128(typemax($Ti))) && (round(x, RoundToZero) == x)
+            if ((!signbit(x) && x <= Float128(typemax($Ti))) && (round(x, RoundToZero) == x)
+                || (x == 0))
                 return unsafe_trunc($Ti,x)
             else
                 throw(InexactError($(Expr(:quote,Ti.name.name)), $Ti, x))
@@ -303,7 +304,8 @@ end
 @assume_effects :foldable round(x::Float128) = Float128(@quad_ccall(libquadmath.rintq(x::Cfloat128)::Cfloat128))
 round(x::Float128, r::RoundingMode{:Down}) = floor(x)
 round(x::Float128, r::RoundingMode{:Up}) = ceil(x)
-round(x::Float128, r::RoundingMode{:ToZero}) = round(x)
+round(x::Float128, r::RoundingMode{:Nearest}) = round(x)
+round(x::Float128, r::RoundingMode{:ToZero}) = trunc(x)
 
 ## two argument
 @assume_effects :foldable (^)(x::Float128, y::Float128) =
@@ -498,8 +500,10 @@ function Float128(x::BigFloat)
         prec = 113 + (k + 16381)
     elseif k == -16381-113 && abs(y) > 0.5
         z = reinterpret(Float128, UInt128(1))
+        return copysign(z,x)
     else
         z = reinterpret(Float128, UInt128(0))
+        return copysign(z,x)
     end
 
     y = BigFloat(x, precision=prec)
diff --git a/test/rounding.jl b/test/rounding.jl
new file mode 100644
index 0000000..7015bcc
--- /dev/null
+++ b/test/rounding.jl
@@ -0,0 +1,310 @@
+# Derived from Julia Base test suite. License is MIT: https://julialang.org/license
+
+# Small sanity tests to ensure changing the rounding of float functions work
+using Base.MathConstants
+
+q1 = one(Float128)
+q0 = zero(Float128)
+
+@testset "Arithmetic rounding checks" begin
+    # a + b returns a number exactly between prevfloat(1.) and 1., so its
+    # final result depends strongly on the utilized rounding direction.
+    a = prevfloat(Float128(0.5))
+    b = Float128(0.5)
+    c = eps(Float128)/4
+    d = prevfloat(Float128(1))
+
+    @testset "Default rounding direction, RoundNearest" begin
+        @test a + b === q1
+        @test - a - b === -q1
+        @test a - b === -c
+        @test b - a === c
+    end
+end
+
+@testset "convert with rounding" begin
+    for v = [sqrt(Float128(2)),-1/Float128(3),nextfloat(q1),prevfloat(q1),nextfloat(-q1),
+             prevfloat(-q1),nextfloat(q0),prevfloat(q0)]
+
+        pn = Float64(v,RoundNearest)
+        @test pn == convert(Float64,v)
+
+        pz = Float64(v,RoundToZero)
+        @test abs(pz) <= abs(v) < nextfloat(abs(pz))
+        @test signbit(pz) == signbit(v)
+
+        pd = Float64(v,RoundDown)
+        @test pd <= v < nextfloat(pd)
+
+        pu = Float64(v,RoundUp)
+        @test prevfloat(pu) < v <= pu
+
+        @test pn == pd || pn == pu
+        @test v > 0 ? pz == pd : pz == pu
+        @test pu - pd == eps(pz)
+    end
+
+    # numbers which are not represented exactly in Float128
+    for T in [Float128,]
+        for v in [sqrt(big(2.0)),-big(1.0)/big(3.0),nextfloat(big(1.0)),
+                  prevfloat(big(1.0)), nextfloat(big(0.0)),prevfloat(big(0.0)),
+                  pi,ℯ,eulergamma,catalan,golden,
+                  typemax(Int128),typemax(UInt128),
+        ]
+            pn = T(v,RoundNearest)
+            @test pn == convert(T,BigFloat(v))
+            pz = T(v,RoundToZero)
+            @test pz == setrounding(()->convert(T,BigFloat(v)), BigFloat, RoundToZero)
+            pd = T(v,RoundDown)
+            if v == prevfloat(big(0.0))
+                @test_broken pd == setrounding(()->convert(T,BigFloat(v)), BigFloat, RoundDown)
+            else
+                @test pd == setrounding(()->convert(T,BigFloat(v)), BigFloat, RoundDown)
+            end
+            pu = T(v,RoundUp)
+            if v == nextfloat(big(0.0))
+                @test_broken pu == setrounding(()->convert(T,BigFloat(v)), BigFloat, RoundUp)
+            else
+                @test pu == setrounding(()->convert(T,BigFloat(v)), BigFloat, RoundUp)
+            end
+            @test pn == pd || pn == pu
+            @test v > 0 ? pz == pd : pz == pu
+            @test isinf(pu) || pu - pd == eps(pz)
+        end
+    end
+end
+
+@testset "rounding properties" for Tf in [Float128,]
+    # these should hold for all u, but we just test the smallest and largest
+    # of each binade
+
+    for i in exponent(floatmin(Tf)):exponent(floatmax(Tf))
+        for u in [ldexp(Tf(1.0), i), -ldexp(Tf(1.0), i),
+                  ldexp(prevfloat(Tf(2.0)), i), -ldexp(prevfloat(Tf(2.0)), i)]
+
+            r = round(u, RoundNearest)
+            if isfinite(u)
+                @test isfinite(r)
+                @test isinteger(r)
+                @test abs(r-u) < 0.5 || abs(r-u) == 0.5 && isinteger(r/2)
+                @test signbit(u) == signbit(r)
+            else
+                @test u === r
+            end
+
+            r = round(u, RoundNearestTiesAway)
+            if isfinite(u)
+                @test isfinite(r)
+                @test isinteger(r)
+                @test abs(r-u) < 0.5 || (r-u) == copysign(0.5,u)
+                @test signbit(u) == signbit(r)
+            else
+                @test u === r
+            end
+
+            r = round(u, RoundNearestTiesUp)
+            if isfinite(u)
+                @test isfinite(r)
+                @test isinteger(r)
+                @test -0.5 < r-u <= 0.5
+                @test signbit(u) == signbit(r)
+            else
+                @test u === r
+            end
+
+            r = round(u, RoundFromZero)
+            if isfinite(u)
+                @test isfinite(r)
+                @test isinteger(r)
+                @test signbit(u) ? (r == floor(u)) : (r == ceil(u))
+                @test signbit(u) == signbit(r)
+            else
+                @test u === r
+            end
+        end
+    end
+end
+
+@testset "rounding difficult values" begin
+    for x = Int128(2)^113-10:Int128(2)^113+10
+        y = Float128(x)
+        i = trunc(Int128,y)
+        @test Int128(trunc(y)) == i
+        @test Int128(round(y)) == i
+        @test Int128(floor(y)) == i
+        @test Int128(ceil(y))  == i
+
+        @test round(Int128,y)       == i
+        @test floor(Int128,y)       == i
+        @test ceil(Int128,y)        == i
+    end
+
+    # rounding vectors
+    let ≈(x,y) = x==y && typeof(x)==typeof(y)
+        for t in [Float128,]
+            # try different vector lengths
+            for n in [0,3,255,256]
+                r = (1:n) .- div(n,2)
+                y = t[x/4 for x in r]
+                @test trunc.(y) ≈ t[div(i,4) for i in r]
+                @test floor.(y) ≈ t[i>>2 for i in r]
+                @test ceil.(y)  ≈ t[(i+3)>>2 for i in r]
+                @test round.(y) ≈ t[(i+1+isodd(i>>2))>>2 for i in r]
+                @test broadcast(x -> round(x, RoundNearestTiesAway), y) ≈ t[(i+1+(i>=0))>>2 for i in r]
+                @test broadcast(x -> round(x, RoundNearestTiesUp), y) ≈ t[(i+2)>>2 for i in r]
+                @test broadcast(x -> round(x, RoundFromZero), y) ≈ t[(i+3*(i>=0))>>2 for i in r]
+            end
+        end
+    end
+
+    @test_throws InexactError round(Int,Float128(Inf))
+    @test_throws InexactError round(Int,Float128(NaN))
+    @test round(Int,Float128(2.5)) == 2
+    @test round(Int,Float128(1.5)) == 2
+    @test round(Int,Float128(-2.5)) == -2
+    @test round(Int,Float128(-1.5)) == -2
+    @test round(Int,Float128(2.5),RoundNearestTiesAway) == 3
+    @test round(Int,Float128(1.5),RoundNearestTiesAway) == 2
+    @test round(Int,Float128(2.5),RoundNearestTiesUp) == 3
+    @test round(Int,Float128(1.5),RoundNearestTiesUp) == 2
+    @test round(Int,Float128(-2.5),RoundNearestTiesAway) == -3
+    @test round(Int,Float128(-1.5),RoundNearestTiesAway) == -2
+    @test round(Int,Float128(-2.5),RoundNearestTiesUp) == -2
+    @test round(Int,Float128(-1.5),RoundNearestTiesUp) == -1
+    @test round(Int,Float128(-1.9)) == -2
+    @test round(Int,nextfloat(q1),RoundFromZero) == 2
+    @test round(Int,-nextfloat(q1),RoundFromZero) == -2
+    @test round(Int,prevfloat(q1),RoundFromZero) == 1
+    @test round(Int,-prevfloat(q1),RoundFromZero) == -1
+    #=
+    FIXME: want analogs
+    @test_throws InexactError round(Int64, 9.223372036854776e18)
+    @test       round(Int64, 9.223372036854775e18) == 9223372036854774784
+    @test_throws InexactError round(Int64, -9.223372036854778e18)
+    @test       round(Int64, -9.223372036854776e18) == typemin(Int64)
+    @test_throws InexactError round(UInt64, 1.8446744073709552e19)
+    @test       round(UInt64, 1.844674407370955e19) == 0xfffffffffffff800
+    =#
+    for Ti in [Int,UInt]
+        for Tf in [Float128,]
+            # Note: these threw InexactError
+            @test round(Ti,Tf(-0.0)) == 0
+            @test round(Ti,Tf(-0.0),RoundNearestTiesAway) == 0
+            @test round(Ti,Tf(-0.0),RoundNearestTiesUp) == 0
+
+            @test round(Ti, Tf(0.5)) == 0
+            @test round(Ti, Tf(0.5), RoundNearestTiesAway) == 1
+            @test round(Ti, Tf(0.5), RoundNearestTiesUp) == 1
+
+            @test round(Ti, prevfloat(Tf(0.5))) == 0
+            @test round(Ti, prevfloat(Tf(0.5)), RoundNearestTiesAway) == 0
+            @test round(Ti, prevfloat(Tf(0.5)), RoundNearestTiesUp) == 0
+
+            @test round(Ti, nextfloat(Tf(0.5))) == 1
+            @test round(Ti, nextfloat(Tf(0.5)), RoundNearestTiesAway) == 1
+            @test round(Ti, nextfloat(Tf(0.5)), RoundNearestTiesUp) == 1
+            # Note: these threw InexactError
+            @test round(Ti, Tf(-0.5)) == 0
+            @test round(Ti, Tf(-0.5), RoundNearestTiesUp) == 0
+
+            @test round(Ti, nextfloat(Tf(-0.5))) == 0
+            @test round(Ti, nextfloat(Tf(-0.5)), RoundNearestTiesAway) == 0
+            @test round(Ti, nextfloat(Tf(-0.5)), RoundNearestTiesUp) == 0
+
+            if Ti <: Signed
+                @test round(Ti, Tf(-0.5), RoundNearestTiesAway) == -1
+                @test round(Ti, prevfloat(Tf(-0.5))) == -1
+                @test round(Ti, prevfloat(Tf(-0.5)), RoundNearestTiesAway) == -1
+                @test round(Ti, prevfloat(Tf(-0.5)), RoundNearestTiesUp) == -1
+            else
+                @test_throws InexactError round(Ti, Tf(-0.5), RoundNearestTiesAway)
+                @test_throws InexactError round(Ti, prevfloat(Tf(-0.5)))
+                @test_throws InexactError round(Ti, prevfloat(Tf(-0.5)), RoundNearestTiesAway)
+                @test_throws InexactError round(Ti, prevfloat(Tf(-0.5)), RoundNearestTiesUp)
+            end
+        end
+    end
+
+    # numbers that can't be rounded by trunc(x+0.5)
+    @test round(Int128, Float128(2.0)^112 + 1) == (UInt128(1) << 112) + 1
+end
+
+# custom rounding and significant-digit ops
+@testset "rounding to digits relative to the decimal point" begin
+    @test round(Float128(pi)) ≈ 3.
+    @test round(Float128(pi), base=10) ≈ 3.
+    @test round(Float128(pi), digits=0) ≈ 3.
+    @test round(Float128(pi), digits=1) ≈ 3.1
+    @test round(Float128(pi), digits=3, base=2) ≈ 3.125
+    @test round(Float128(pi), sigdigits=1) ≈ 3.
+    @test round(Float128(pi), sigdigits=3) ≈ 3.14
+    @test round(Float128(pi), sigdigits=4, base=2) ≈ 3.25
+    @test round(10*Float128(pi), digits=-1) ≈ 30.
+    @test round(Float128(.1), digits=0) == 0.
+    @test round(Float128(-.1), digits=0) == -0.
+    @test isnan(round(Float128(NaN), digits=2))
+    @test isinf(round(Float128(Inf), digits=2))
+    @test isinf(round(Float128(-Inf), digits=2))
+end
+@testset "round vs trunc vs floor vs ceil" begin
+    @test round(Float128(123.456), digits=1) ≈ 123.5
+    @test round(-Float128(123.456), digits=1) ≈ -123.5
+    @test trunc(Float128(123.456), digits=1) ≈ 123.4
+    @test trunc(-Float128(123.456), digits=1) ≈ -123.4
+    @test ceil(Float128(123.456), digits=1) ≈ 123.5
+    @test ceil(-Float128(123.456), digits=1) ≈ -123.4
+    @test floor(Float128(123.456), digits=1) ≈ 123.4
+    @test floor(-Float128(123.456), digits=1) ≈ -123.5
+end
+@testset "rounding with too much (or too few) precision" begin
+    for x in (12345.6789, 0, -12345.6789)
+        y = Float128(x)
+        @test y == trunc(x, digits=1000)
+        @test y == round(x, digits=1000)
+        @test y == floor(x, digits=1000)
+        @test y == ceil(x, digits=1000)
+    end
+    let x = Float128(12345.6789)
+        @test 0.0 == trunc(x, digits=-1000)
+        @test 0.0 == round(x, digits=-1000)
+        @test 0.0 == floor(x, digits=-1000)
+        @test Inf128 == ceil(x, digits=-10000)
+    end
+    let x = Float128(-12345.6789)
+        @test -0.0 == trunc(x, digits=-1000)
+        @test -0.0 == round(x, digits=-1000)
+        @test -Inf128 == floor(x, digits=-10000)
+        @test -0.0 == ceil(x, digits=-1000)
+    end
+    let x = q0
+        @test 0.0 == trunc(x, digits=-1000)
+        @test 0.0 == round(x, digits=-1000)
+        @test 0.0 == floor(x, digits=-1000)
+        @test 0.0 == ceil(x, digits=-1000)
+    end
+end
+@testset "rounding in other bases" begin
+    @test round(Float128(pi), digits = 2, base = 2) ≈ 3.25
+    @test round(Float128(pi), digits = 3, base = 2) ≈ 3.125
+    @test round(Float128(pi), digits = 3, base = 5) ≈ 3.144
+end
+
+if VERSION >= v"1.11"
+@testset "rounding floats with specified return type #50778" begin
+    @test round(Float128, Float128(1.2)) === q1
+end
+
+# The libmpfr shipped with Julia is built without float128 support, so we can't
+# easily do comparisons to conversion and rounding by library calls.
+# If this situation changes, consider adapting more tests from Base.
+
+@testset "floor(<:AbstractFloat, large_number) (#52355)" begin
+    for i in [-BigInt(floatmax(Float128)), -BigInt(floatmax(Float128))*100,
+              BigInt(floatmax(Float128)), BigInt(floatmax(Float128))*100]
+        f = ceil(Float128, i)
+        @test f >= i
+        @test isinteger(f) || isinf(f)
+        @test prevfloat(f) < i
+    end
+end
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index cc85c1b..49c81ac 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -273,5 +273,7 @@ include("specfun.jl")
 
 include("printf.jl")
 
+@testset verbose=true "rounding" include("rounding.jl")
+
 using Aqua
 Aqua.test_all(Quadmath)