diff --git a/Project.toml b/Project.toml
index 13264cc3..f4ee73e4 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "FillArrays"
 uuid = "1a297f60-69ca-5386-bcde-b61e274b549b"
-version = "1.7.0"
+version = "1.7.1"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
diff --git a/src/FillArrays.jl b/src/FillArrays.jl
index 7ca7b78e..23aa6279 100644
--- a/src/FillArrays.jl
+++ b/src/FillArrays.jl
@@ -550,16 +550,6 @@ for SMT in (:Diagonal, :Bidiagonal, :Tridiagonal, :SymTridiagonal)
     end
 end
 
-
-#########
-# maximum/minimum
-#########
-
-for op in (:maximum, :minimum)
-    @eval $op(x::AbstractFill) = getindex_value(x)
-end
-
-
 #########
 # Cumsum
 #########
diff --git a/src/fillbroadcast.jl b/src/fillbroadcast.jl
index 6cf11284..36c4ac3d 100644
--- a/src/fillbroadcast.jl
+++ b/src/fillbroadcast.jl
@@ -33,22 +33,81 @@ end
 ### mapreduce
 
 function Base._mapreduce_dim(f, op, ::Base._InitialValue, A::AbstractFill, ::Colon)
-    fval = f(getindex_value(A))
-    out = fval
-    for _ in 2:length(A)
-        out = op(out, fval)
+    if length(A) == 0
+        return Base.mapreduce_empty_iter(f, op, A, Base.HasEltype())
+    end
+    val = getindex_value(A)
+    out = Base.mapreduce_first(f, op, val)
+    fval = f(val)
+    if op(out, fval) != out
+        for _ in 2:length(A)
+            out = op(out, fval)
+        end
+    end
+    out
+end
+
+function Base._mapreduce_dim(f, op, init, A::AbstractFill, ::Colon)
+    if length(A) == 0
+        return init
+    end
+    val = getindex_value(A)
+    fval = f(val)
+    out = op(init, fval)
+    if op(out, fval) != out
+        for _ in 2:length(A)
+            out = op(out, fval)
+        end
     end
     out
 end
 
+identityel(f, ::Union{typeof(+), typeof(Base.add_sum)}, A) = zero(f(zero(eltype(A))))
+identityel(f, ::Union{typeof(*), typeof(Base.mul_prod)}, A) = one(f(one(eltype(A))))
+identityel(f, ::typeof(&), A) = true
+identityel(f, ::typeof(|), A) = false
+identityel(f, ::Any, @nospecialize(A)) = throw(ArgumentError("reducing over an empty collection is not allowed"))
+function mapreducedim_empty(f, op, A)
+    z = identityel(f, op, A)
+    op(z, z)
+end
+
+function reduced_indices(A, dims)
+    ntuple(d -> d in dims ? axes(A,ndims(A)+1) : axes(A,d), ndims(A))
+end
+
 function Base._mapreduce_dim(f, op, ::Base._InitialValue, A::AbstractFill, dims)
-    fval = f(getindex_value(A))
     red = *(ntuple(d -> d in dims ? size(A,d) : 1, ndims(A))...)
-    out = fval
-    for _ in 2:red
-        out = op(out, fval)
+    if red == 0
+        out = mapreducedim_empty(f, op, A)
+    else
+        val = getindex_value(A)
+        out = Base.mapreduce_first(f, op, val)
+        fval = f(val)
+        if op(out, fval) != out
+            for _ in 2:red
+                out = op(out, fval)
+            end
+        end
+    end
+    Fill(out, reduced_indices(A, dims))
+end
+
+function Base._mapreduce_dim(f, op, init, A::AbstractFill, dims)
+    red = *(ntuple(d -> d in dims ? size(A,d) : 1, ndims(A))...)
+    if red == 0
+        out = init
+    else
+        val = getindex_value(A)
+        fval = f(val)
+        out = op(init, fval)
+        if op(out, fval) != out
+            for _ in 2:red
+                out = op(out, fval)
+            end
+        end
     end
-    Fill(out, ntuple(d -> d in dims ? Base.OneTo(1) : axes(A,d), ndims(A)))
+    Fill(out, reduced_indices(A, dims))
 end
 
 function mapreduce(f, op, A::AbstractFill, B::AbstractFill; kw...)
diff --git a/test/runtests.jl b/test/runtests.jl
index 50d42c54..2bacc8c6 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -1090,7 +1090,7 @@ end
                 @test Zeros(S, 10) .* (T(1):T(10)) ≡ Zeros(U, 10)
                 @test_throws DimensionMismatch Zeros(S, 10) .* (T(1):T(11))
             end
-        end        
+        end
     end
 end
 
@@ -1115,6 +1115,62 @@ end
 end
 
 @testset "mapreduce" begin
+    @testset "corner cases with small arrays" begin
+        @test_throws Exception reduce(max, Fill(2,0))
+        @test_throws Exception reduce(max, Fill(2,0), dims=1)
+        function testreduce(op, A; kw...)
+            B = Array(A)
+            F = reduce(op, A; kw...)
+            @test F == reduce(op, B; kw...)
+            if haskey(kw, :dims)
+                @test F isa Fill
+            end
+            if !isempty(A)
+                @test mapreduce(x->x^2, op, A; kw...) == mapreduce(x->x^2, op, B; kw...)
+            end
+        end
+        @testset for (op, init) in ((+, 0), (*, 1))
+            testreduce(op, Fill(2,0))
+            testreduce(op, Fill(2,0); init)
+            testreduce(op, Fill(2,0), dims=1)
+            testreduce(op, Fill(2,0); init, dims=1)
+        end
+        @testset for (op, init) in ((&, true), (|, false))
+            testreduce(op, Fill(true,0))
+            testreduce(op, Fill(true,0); init)
+            testreduce(op, Fill(true,0), dims=1)
+            testreduce(op, Fill(true,0); init, dims=1)
+        end
+        testreduce(vcat, Fill(2,0), init=Int[])
+        testreduce(vcat, Fill(2,1), init=Int[])
+        @testset for (op, init) in ((max, 0), (+, 0), (*, 1))
+            testreduce(op, Fill(2,0), dims=2)
+            testreduce(op, Fill(2,0,1), dims=2)
+            testreduce(op, Fill(2,0); init, dims=2)
+            testreduce(op, Fill(2,0,1); init, dims=2)
+
+            testreduce(op, Fill(2,1))
+            testreduce(op, Fill(2,1), dims=1)
+            testreduce(op, Fill(2,1), dims=2)
+            testreduce(op, Fill(2,1,1), dims=1)
+            testreduce(op, Fill(2,1,1), dims=2)
+            testreduce(op, Fill(2,1); init, dims=1)
+            testreduce(op, Fill(2,1); init, dims=2)
+            testreduce(op, Fill(2,1,1); init, dims=1)
+            testreduce(op, Fill(2,1,1); init, dims=2)
+
+            testreduce(op, Fill(2,2))
+            testreduce(op, Fill(2,2), dims=1)
+            testreduce(op, Fill(2,2), dims=2)
+            testreduce(op, Fill(2,2,2), dims=1)
+            testreduce(op, Fill(2,2,2), dims=2)
+            testreduce(op, Fill(2,2); init, dims=1)
+            testreduce(op, Fill(2,2); init, dims=2)
+            testreduce(op, Fill(2,2,2); init, dims=1)
+            testreduce(op, Fill(2,2,2); init, dims=2)
+        end
+    end
+
     x = rand(3, 4)
     y = fill(1.0, 3, 4)
     Y = Fill(1.0, 3, 4)