diff --git a/src/broadcast.jl b/src/broadcast.jl
index db74f89e..dd4be227 100644
--- a/src/broadcast.jl
+++ b/src/broadcast.jl
@@ -97,11 +97,15 @@ scalar_getindex(x::Ref) = x[]
 scalar_getindex(x::Tuple{<: Any}) = x[1]
 
 @generated function _broadcast(f, ::Size{newsize}, s::Tuple{Vararg{Size}}, a...) where newsize
-    first_staticarray = 0
-    for i = 1:length(a)
-        if a[i] <: StaticArray
-            first_staticarray = a[i]
-            break
+    first_staticarray = a[findfirst(ai -> ai <: StaticArray, a)]
+
+    if prod(newsize) == 0
+        # Use inference to get eltype in empty case (see also comments in _map)
+        eltys = [:(eltype(a[$i])) for i ∈ 1:length(a)]
+        return quote
+            @_inline_meta
+            T = Core.Compiler.return_type(f, Tuple{$(eltys...)})
+            @inbounds return similar_type($first_staticarray, T, Size(newsize))()
         end
     end
 
diff --git a/src/mapreduce.jl b/src/mapreduce.jl
index 7806bba3..6b5c80f1 100644
--- a/src/mapreduce.jl
+++ b/src/mapreduce.jl
@@ -18,17 +18,31 @@ end
 end
 
 @generated function _map(f, a::AbstractArray...)
-    i = findfirst(ai -> ai <: StaticArray, a)
-    if i === nothing
+    first_staticarray = findfirst(ai -> ai <: StaticArray, a)
+    if first_staticarray === nothing
         return :(throw(ArgumentError("No StaticArray found in argument list")))
     end
     # Passing the Size as an argument to _map leads to inference issues when
     # recursively mapping over nested StaticArrays (see issue #593). Calling
-    # Size in the generator here is valid because a[i] is known to be a
+    # Size in the generator here is valid because a[first_staticarray] is known to be a
     # StaticArray for which the default Size method is correct. If wrapped
     # StaticArrays (with a custom Size method) are to be supported, this will
     # no longer be valid.
-    S = Size(a[i])
+    S = Size(a[first_staticarray])
+
+    if prod(S) == 0
+        # In the empty case only, use inference to try figuring out a sensible
+        # eltype, as is done in Base.collect and broadcast.
+        # See https://github.com/JuliaArrays/StaticArrays.jl/issues/528
+        eltys = [:(eltype(a[$i])) for i ∈ 1:length(a)]
+        return quote
+            @_inline_meta
+            S = same_size(a...)
+            T = Core.Compiler.return_type(f, Tuple{$(eltys...)})
+            @inbounds return similar_type(a[$first_staticarray], T, S)()
+        end
+    end
+
     exprs = Vector{Expr}(undef, prod(S))
     for i ∈ 1:prod(S)
         tmp = [:(a[$j][$i]) for j ∈ 1:length(a)]
diff --git a/test/broadcast.jl b/test/broadcast.jl
index a6209d5a..eebce977 100644
--- a/test/broadcast.jl
+++ b/test/broadcast.jl
@@ -109,9 +109,6 @@ end
         @test @inferred(v2 .- v1) === SVector(0, 2)
         @test @inferred(v1 .^ v2) === SVector(1, 16)
         @test @inferred(v2 .^ v1) === SVector(1, 16)
-        # Issue #199: broadcast with empty SArray
-        @test @inferred(SVector(1) .+ SVector{0,Int}()) === SVector{0,Union{}}()
-        @test @inferred(SVector{0,Int}() .+ SVector(1)) === SVector{0,Union{}}()
         # Issue #200: broadcast with Adjoint
         @test @inferred(v1 .+ v2') === @SMatrix [2 5; 3 6]
         @test @inferred(v1 .+ transpose(v2)) === @SMatrix [2 5; 3 6]
@@ -142,6 +139,13 @@ end
         @test @inferred(zeros(SVector{0}) .+ zeros(SMatrix{0,2})) === zeros(SMatrix{0,2})
         m = zeros(MMatrix{0,2})
         @test @inferred(broadcast!(+, m, m, zeros(SVector{0}))) == zeros(SMatrix{0,2})
+        # Issue #199: broadcast with empty SArray
+        @test @inferred(SVector(1) .+ SVector{0,Int}()) === SVector{0,Int}()
+        @test @inferred(SVector{0,Int}() .+ SVector(1.0)) === SVector{0,Float64}()
+        # Issue #528
+        @test @inferred(isapprox(SMatrix{3,0,Float64}(), SMatrix{3,0,Float64}()))
+        @test @inferred(broadcast(length, SVector{0,String}())) === SVector{0,Int}()
+        @test @inferred(broadcast(join, SVector{0,String}(), SVector{0,String}(), SVector{0,String}())) === SVector{0,String}()
     end
 
     @testset "Mutating broadcast!" begin
diff --git a/test/linalg.jl b/test/linalg.jl
index ad7288ad..6b070827 100644
--- a/test/linalg.jl
+++ b/test/linalg.jl
@@ -2,7 +2,7 @@ using StaticArrays, Test, LinearAlgebra
 
 @testset "Linear algebra" begin
 
-    @testset "SVector as a (mathematical) vector space" begin
+    @testset "SArray as a (mathematical) vector space" begin
         c = 2
         v1 = @SVector [2,4,6,8]
         v2 = @SVector [4,3,2,1]
@@ -14,6 +14,10 @@ using StaticArrays, Test, LinearAlgebra
         @test @inferred(v1 + v2) === @SVector [6, 7, 8, 9]
         @test @inferred(v1 - v2) === @SVector [-2, 1, 4, 7]
 
+        # #528 eltype with empty addition
+        zm = zeros(SMatrix{3, 0, Float64})
+        @test @inferred(zm + zm) === zm
+
         # TODO Decide what to do about this stuff:
         #v3 = [2,4,6,8]
         #v4 = [4,3,2,1]
diff --git a/test/lu.jl b/test/lu.jl
index a1375698..6c009cac 100644
--- a/test/lu.jl
+++ b/test/lu.jl
@@ -36,9 +36,5 @@ using StaticArrays, Test, LinearAlgebra
 
     # decomposition is correct
     l_u = l*u
-    if length(l_u) > 0 # Union{} element type breaks norm
-        @test l*u ≈ a[p,:]
-    else
-        @test_broken l*u ≈ a[p,:]
-    end
+    @test l*u ≈ a[p,:]
 end
diff --git a/test/mapreduce.jl b/test/mapreduce.jl
index e76e6f82..084a3ee7 100644
--- a/test/mapreduce.jl
+++ b/test/mapreduce.jl
@@ -24,6 +24,11 @@ using Statistics: mean
         v3 = @SVector [1, 2, 3, 4]
         map!(+, mv3, v1, v2, v3)
         @test mv3 == @MVector [7, 9, 11, 13]
+
+        # Output eltype for empty cases #528
+        @test @inferred(map(/, SVector{0,Int}(), SVector{0,Int}())) === SVector{0,Float64}()
+        @test @inferred(map(+, SVector{0,Int}(), SVector{0,Float32}())) === SVector{0,Float32}()
+        @test @inferred(map(length, SVector{0,String}())) === SVector{0,Int}()
     end
 
     @testset "[map]reduce and [map]reducedim" begin