diff --git a/Project.toml b/Project.toml
index 17c28a38..0a5c562e 100644
--- a/Project.toml
+++ b/Project.toml
@@ -9,10 +9,12 @@ LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 PDMats = "90014a1f-27ba-587c-ab20-58faa44d9150"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
+StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 
 [extensions]
 FillArraysPDMatsExt = "PDMats"
 FillArraysSparseArraysExt = "SparseArrays"
+FillArraysStaticArraysExt = "StaticArrays"
 FillArraysStatisticsExt = "Statistics"
 
 [compat]
diff --git a/ext/FillArraysSparseArraysExt.jl b/ext/FillArraysSparseArraysExt.jl
index 4cb29dcd..68bfc5dd 100644
--- a/ext/FillArraysSparseArraysExt.jl
+++ b/ext/FillArraysSparseArraysExt.jl
@@ -4,7 +4,7 @@ using SparseArrays
 using SparseArrays: SparseVectorUnion
 import Base: convert, kron
 using FillArrays
-using FillArrays: RectDiagonalFill, RectOrDiagonalFill, ZerosVector, ZerosMatrix, getindex_value, AbstractFillVector, _fill_dot
+using FillArrays: RectDiagonalFill, RectOrDiagonalFill, ZerosVector, ZerosMatrix, getindex_value, AbstractFillVector, _fill_dot, OneElementVector, OneElementMatrix
 # Specifying the full namespace is necessary because of https://github.com/JuliaLang/julia/issues/48533
 # See https://github.com/JuliaStats/LogExpFunctions.jl/pull/63
 using FillArrays.LinearAlgebra
@@ -63,4 +63,16 @@ end
 # Ambiguity. see #178
 dot(x::AbstractFillVector, y::SparseVectorUnion) = _fill_dot(x, y)
 
+# OneElements with different indices should return
+# SparseArrays under addition
+function FillArrays.oneelement_addsub(a::OneElementVector, b::OneElementVector, aval, bval)
+    return sparsevec([a.ind[1], b.ind[1]], [aval, bval], length(a))
+end
+
+function FillArrays.oneelement_addsub(a::OneElementMatrix, b::OneElementMatrix, aval, bval)
+    nzval = [aval, bval]
+    nzind = ntuple(i -> [a.ind[i], b.ind[i]], ndims(a))
+    return sparse(nzind..., nzval, size(a)...)
+end
+
 end # module
diff --git a/ext/FillArraysStaticArraysExt.jl b/ext/FillArraysStaticArraysExt.jl
new file mode 100644
index 00000000..f120be1a
--- /dev/null
+++ b/ext/FillArraysStaticArraysExt.jl
@@ -0,0 +1,28 @@
+module FillArraysStaticArraysExt
+
+using FillArrays
+using StaticArrays
+
+import Base: promote_op
+import FillArrays: elconvert
+
+# Disambiguity methods for StaticArrays
+
+function Base.:+(a::FillArrays.Zeros, b::StaticArray)
+    promote_shape(a,b)
+    return elconvert(promote_op(+,eltype(a),eltype(b)),b)
+end
+function Base.:+(a::StaticArray, b::FillArrays.Zeros)
+    promote_shape(a,b)
+    return elconvert(promote_op(+,eltype(a),eltype(b)),a)
+end
+function Base.:-(a::StaticArray, b::FillArrays.Zeros)
+    promote_shape(a,b)
+    return elconvert(promote_op(-,eltype(a),eltype(b)),a)
+end
+function Base.:-(a::FillArrays.Zeros, b::StaticArray)
+    promote_shape(a,b)
+    return elconvert(promote_op(-,eltype(a),eltype(b)),-b)
+end
+
+end # module
diff --git a/src/fillalgebra.jl b/src/fillalgebra.jl
index deb7e74d..62e390ef 100644
--- a/src/fillalgebra.jl
+++ b/src/fillalgebra.jl
@@ -436,16 +436,29 @@ function +(a::AbstractZeros{T}, b::AbstractZeros{V}) where {T, V} # for disambig
     promote_shape(a,b)
     return elconvert(promote_op(+,T,V),a)
 end
-# no AbstractArray. Otherwise incompatible with StaticArrays.jl
-# AbstractFill for disambiguity
-for TYPE in (:Array, :AbstractFill, :AbstractRange, :Diagonal)
+
+# AbstractFill and Array for disambiguity
+for TYPE in (:Array, :AbstractFill, :AbstractRange, :AbstractArray)
     @eval function +(a::$TYPE{T}, b::AbstractZeros{V}) where {T, V}
         promote_shape(a,b)
         return elconvert(promote_op(+,T,V),a)
     end
+    @eval function -(a::$TYPE{T}, b::AbstractZeros{V}) where {T, V}
+        promote_shape(a,b)
+        return elconvert(promote_op(-,T,V),a)
+    end
+    @eval function -(a::AbstractZeros{T}, b::$TYPE{V}) where {T, V}
+        promote_shape(a,b)
+        return elconvert(promote_op(-,T,V),-b)
+    end
     @eval +(a::AbstractZeros, b::$TYPE) = b + a
 end
 
+function -(a::AbstractZeros, b::AbstractZeros)
+    promote_shape(a,b)
+    return elconvert(promote_type(eltype(a), eltype(b)),-b)
+end
+
 # for VERSION other than 1.6, could use ZerosMatrix only
 function +(a::AbstractFillMatrix{T}, b::UniformScaling) where {T}
     n = checksquare(a)
diff --git a/src/oneelement.jl b/src/oneelement.jl
index f2db8565..dabfbf06 100644
--- a/src/oneelement.jl
+++ b/src/oneelement.jl
@@ -148,6 +148,90 @@ end
 /(x::OneElement, b::Number) = OneElement(x.val / b, x.ind, x.axes)
 \(b::Number, x::OneElement) = OneElement(b \ x.val, x.ind, x.axes)
 
+# Addition
+
+# O(1) addition with arbitrary array types
+function add_one_elem(a::OneElement, b::AbstractArray)
+    axes(a) == axes(b) || throw(DimensionMismatch(LazyString("A has dimensions ", size(a), " but B has dimensions ", size(b))))
+
+    ret = copy(b)
+    try
+        ret[a.ind...] += getindex_value(a)
+    catch
+        # Fallback to materialising dense array if setindex!
+        # goes wrong (e.g on a Diagonal)
+        ret = Array(ret)
+        checkbounds(Bool, ret, a.ind...) && (ret[a.ind...] += getindex_value(a))
+    end
+    return ret
+end
+
+function sub_one_elem(a::AbstractArray, b::OneElement)
+    axes(a) == axes(b) || throw(DimensionMismatch(LazyString("A has dimensions ", size(a), " but B has dimensions ", size(b))))
+    ret = copy(a)
+    try
+        ret[b.ind...] -= getindex_value(b)
+    catch
+        # Fallback to materialising dense array if setindex!
+        # goes wrong (e.g on a Diagonal)
+        ret = Array(ret)
+        checkbounds(Bool, ret, b.ind...) && (ret[b.ind...] -= getindex_value(b))
+    end
+    return ret
+end
+
+function sub_one_elem(a::OneElement, b::AbstractArray)
+    axes(a) == axes(b) || throw(DimensionMismatch(LazyString("A has dimensions ", size(a), " but B has dimensions ", size(b))))
+    ret = copy(-b)
+    try
+        ret[a.ind...] += getindex_value(a)
+    catch
+        # Fallback to materialising dense array if setindex!
+        # goes wrong (e.g on a Diagonal)
+        ret = Array(ret)
+        checkbounds(Bool, ret, a.ind...) && (ret[a.ind...] += getindex_value(a))
+    end
+    return ret
+end
+
++(a::OneElement, b::AbstractArray) = add_one_elem(a, b)
++(a::AbstractArray, b::OneElement) = add_one_elem(b, a)
+-(a::AbstractArray, b::OneElement) = sub_one_elem(a, b)
+-(a::OneElement, b::AbstractArray) = sub_one_elem(a, b)
+# disambiguity
+function +(a::AbstractZeros, b::OneElement)
+    promote_shape(a,b)
+    return elconvert(promote_op(+,eltype(a),eltype(b)),b)
+end
+
++(a::OneElement, b::AbstractZeros) = b + a
+
+# Adding/subtracting OneElements
+# (Without SparseArrays) materialise dense vector if indices are different
+
+# Sparse Arrays extension overrides this for OneElementVector and OneElementMatrix
+function oneelement_addsub(a::OneElement, b::OneElement, aval, bval)
+    ret = similar(a)
+    fill!(ret, zero(eltype(ret)))
+    ret[a.ind...] = aval
+    ret[b.ind...] = bval
+    return ret
+end
+
+for (op, bop) in (:+ => :(getindex_value(b)),
+                  :- => :(-getindex_value(b)))
+    @eval begin
+        function $op(a::OneElement, b::OneElement)
+            axes(a) == axes(b) || throw(DimensionMismatch(LazyString("A has dimensions ", size(a), " but B has dimensions ", size(b))))
+            if a.ind == b.ind
+                return OneElement($op(getindex_value(a), getindex_value(b)), a.ind, axes(a))
+            else
+                return oneelement_addsub(a, b, getindex_value(a), $bop)
+            end
+        end
+    end
+end
+
 # matrix-vector and matrix-matrix multiplication
 
 # Fill and OneElement
diff --git a/test/runtests.jl b/test/runtests.jl
index 174075cc..7ab9585a 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -505,6 +505,14 @@ end
         for A in As, Z in (TZ -> Zeros{TZ}(3)).((Int, Float64, Int8, ComplexF64))
             test_addition_and_subtraction_dim_mismatch(A, Z)
         end
+
+        # Zeros should be additive identity for matrices
+        D = Diagonal([1, 1])
+        Z = Zeros(2, 2)
+        @test D + Z isa Diagonal
+        @test D + Z == D
+        @test D - Z == D
+        @test Z - D == -D
     end
 end
 
@@ -588,6 +596,12 @@ end
             testsparsediag(E)
         end
     end
+
+    # Adding one elements with different indices should return a sparse array.
+    A = OneElement(2, 5)
+    B = OneElement(3, 5)
+    @test A + B isa SparseVector
+    @test A + B == [0, 1, 1, 0, 0]
 end
 
 @testset "==" begin
@@ -2374,6 +2388,19 @@ end
         @test_throws ArgumentError isassigned(f, true)
     end
 
+    @testset "Addition/Subtraction" begin
+        A = OneElement(2, 5)
+        B = OneElement(3, 5)
+
+        @test A + A isa OneElement
+        @test A + A == OneElement(2, 2, 5)
+        @test A + B == [0, 1, 1, 0, 0]
+
+        @test B - B isa OneElement
+        @test B - B == OneElement(0, 2, 5)
+        @test B - A == [0, -1, 1, 0, 0]
+    end
+
     @testset "matmul" begin
         A = reshape(Float64[1:9;], 3, 3)
         v = reshape(Float64[1:3;], 3)