diff --git a/.gitignore b/.gitignore
new file mode 100644
index 00000000..ba39cc53
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1 @@
+Manifest.toml
diff --git a/src/LinearMaps.jl b/src/LinearMaps.jl
index bd030de8..45de6023 100644
--- a/src/LinearMaps.jl
+++ b/src/LinearMaps.jl
@@ -130,7 +130,7 @@ julia> A(x)
 """
 function Base.:(*)(A::LinearMap, x::AbstractVector)
     check_dim_mul(A, x)
-    T = promote_type(eltype(A), eltype(x))
+    T = typeof(oneunit(eltype(A)) * oneunit(eltype(x)))
     y = similar(x, T, axes(A)[1])
     return @inbounds mul!(y, A, x)
 end
@@ -308,25 +308,23 @@ function _generic_map_mul!(Y, A, X::AbstractMatrix, α, β)
     end
     return Y
 end
-function _generic_map_mul!(Y, A, s::Number)
-    T = promote_type(eltype(A), typeof(s))
+function _generic_map_mul!(Y, A, s::S) where {S <: Number}
     ax2 = axes(A)[2]
-    xi = zeros(T, ax2)
+    xi = zeros(S, ax2)
     @inbounds for (i, Yi) in zip(ax2, eachcol(Y))
         xi[i] = s
         mul!(Yi, A, xi)
-        xi[i] = zero(T)
+        xi[i] = zero(S)
     end
     return Y
 end
-function _generic_map_mul!(Y, A, s::Number, α, β)
-    T = promote_type(eltype(A), typeof(s))
+function _generic_map_mul!(Y, A, s::S, α, β) where {S <: Number}
     ax2 = axes(A)[2]
-    xi = zeros(T, ax2)
+    xi = zeros(S, ax2)
     @inbounds for (i, Yi) in zip(ax2, eachcol(Y))
         xi[i] = s
         mul!(Yi, A, xi, α, β)
-        xi[i] = zero(T)
+        xi[i] = zero(S)
     end
     return Y
 end
diff --git a/src/composition.jl b/src/composition.jl
index fe594c17..4717815d 100644
--- a/src/composition.jl
+++ b/src/composition.jl
@@ -1,3 +1,9 @@
+# appropriate type for product of two (possibly Unitful) quantities:
+_multype(a, b) = typeof(oneunit(eltype(a)) * oneunit(eltype(b)))
+# this variant is needed because reinterpret changes length of complex vectors
+#_multype(a, b, iscomplex::Bool) =
+#    typeof((1+0im) * oneunit(eltype(a)) * oneunit(eltype(b)))
+
 struct CompositeMap{T, As<:LinearMapTupleOrVector} <: LinearMap{T}
     maps::As # stored in order of application to vector
     function CompositeMap{T, As}(maps::As) where {T, As}
@@ -5,17 +11,19 @@ struct CompositeMap{T, As<:LinearMapTupleOrVector} <: LinearMap{T}
         for n in 2:N
             check_dim_mul(maps[n], maps[n-1])
         end
-        for TA in Base.Iterators.map(eltype, maps)
-            promote_type(T, TA) == T ||
-                error("eltype $TA cannot be promoted to $T in CompositeMap constructor")
-        end
+        Tprod = typeof(*(map(oneunit∘eltype, maps)...)) # handles units
+        promote_type(T, Tprod) == T ||
+            error("eltype $Tprod and $T incompatible in CompositeMap constructor")
         new{T, As}(maps)
     end
 end
+
 CompositeMap{T}(maps::As) where {T, As<:LinearMapTupleOrVector} = CompositeMap{T, As}(maps)
 
-Base.mapreduce(::typeof(identity), ::typeof(Base.mul_prod), maps::LinearMapTupleOrVector) =
-    CompositeMap{promote_type(map(eltype, maps)...)}(reverse(maps))
+function Base.mapreduce(::typeof(identity), ::typeof(Base.mul_prod), maps::LinearMapTupleOrVector)
+    Tprod = typeof(*(map(oneunit∘eltype, maps)...)) # handles units
+    return CompositeMap{Tprod}(reverse(maps))
+end
 Base.mapreduce(::typeof(identity), ::typeof(Base.mul_prod), maps::AbstractVector{<:LinearMap{T}}) where {T} =
     CompositeMap{T}(reverse(maps))
 
@@ -80,11 +88,11 @@ end
 
 # scalar multiplication and division (non-commutative case)
 function Base.:(*)(α::Number, A::LinearMap)
-    T = promote_type(typeof(α), eltype(A))
+    T = _multype(α, A)
     return CompositeMap{T}(_combine(A, UniformScalingMap(α, size(A, 1))))
 end
 function Base.:(*)(α::Number, A::CompositeMap)
-    T = promote_type(typeof(α), eltype(A))
+    T = _multype(α, A)
     Alast = last(A.maps)
     if Alast isa UniformScalingMap
         return CompositeMap{T}(_combine(_front(A.maps), α * Alast))
@@ -94,15 +102,15 @@ function Base.:(*)(α::Number, A::CompositeMap)
 end
 # needed for disambiguation
 function Base.:(*)(α::RealOrComplex, A::CompositeMap{<:RealOrComplex})
-    T = Base.promote_op(*, typeof(α), eltype(A))
+    T = _multype(α, A)
     return ScaledMap{T}(α, A)
 end
 function Base.:(*)(A::LinearMap, α::Number)
-    T = promote_type(typeof(α), eltype(A))
+    T = _multype(A, α)
     return CompositeMap{T}(_combine(UniformScalingMap(α, size(A, 2)), A))
 end
 function Base.:(*)(A::CompositeMap, α::Number)
-    T = promote_type(typeof(α), eltype(A))
+    T = _multype(A, α)
     Afirst = first(A.maps)
     if Afirst isa UniformScalingMap
         return CompositeMap{T}(_combine(Afirst * α, _tail(A.maps)))
@@ -112,7 +120,7 @@ function Base.:(*)(A::CompositeMap, α::Number)
 end
 # needed for disambiguation
 function Base.:(*)(A::CompositeMap{<:RealOrComplex}, α::RealOrComplex)
-    T = Base.promote_op(*, typeof(α), eltype(A))
+    T = _multype(A, α)
     return ScaledMap{T}(α, A)
 end
 
@@ -137,19 +145,19 @@ julia> LinearMap(ones(Int, 3, 3)) * CS * I * rand(3, 3);
 ```
 """
 function Base.:(*)(A₁::LinearMap, A₂::LinearMap)
-    T = promote_type(eltype(A₁), eltype(A₂))
+    T = _multype(A₁, A₂)
     return CompositeMap{T}(_combine(A₂, A₁))
 end
 function Base.:(*)(A₁::LinearMap, A₂::CompositeMap)
-    T = promote_type(eltype(A₁), eltype(A₂))
+    T = _multype(A₁, A₂)
     return CompositeMap{T}(_combine(A₂.maps, A₁))
 end
 function Base.:(*)(A₁::CompositeMap, A₂::LinearMap)
-    T = promote_type(eltype(A₁), eltype(A₂))
+    T = _multype(A₁, A₂)
     return CompositeMap{T}(_combine(A₂, A₁.maps))
 end
 function Base.:(*)(A₁::CompositeMap, A₂::CompositeMap)
-    T = promote_type(eltype(A₁), eltype(A₂))
+    T = _multype(A₁, A₂)
     return CompositeMap{T}(_combine(A₂.maps, A₁.maps))
 end
 # needed for disambiguation
@@ -217,6 +225,11 @@ function _compositemulN!(y, A::CompositeMap, x,
                          src = nothing,
                          dst = nothing)
     N = length(A.maps) # ≥ 3
+    # caution: be careful if y is complex but intermediate products are not
+    #   source = reinterpret(_multype(A.maps[1], x, !isreal(y)), source) # for units
+    #   resize!(source, size(A.maps[1],1)) # trick due to complex case
+    # todo: build reinterpret into _unsafe_mul! instead?
+    # only necessary if either source or map has Number type instead of Real|Complex
     n = n0 = firstindex(A.maps)
     source = isnothing(src) ?
         convert(AbstractArray, A.maps[n] * x) :
@@ -227,9 +240,12 @@ function _compositemulN!(y, A::CompositeMap, x,
         _unsafe_mul!(dst, A.maps[n], source)
     dest, source = source, dest # alternate dest and source
     for n in (n0+2):N-1
-        dest = _resize(dest, (size(A.maps[n], 1), size(x)[2:end]...))
+        # dest = _resize(dest, (size(A.maps[n], 1), size(x)[2:end]...))
+        # dest = reinterpret(_multype(A.maps[n], source), dest)
+        dest = similar([], _multype(A.maps[n], source), (size(A.maps[n], 1), size(source)[2:end]...))
         _unsafe_mul!(dest, A.maps[n], source)
-        dest, source = source, dest # alternate dest and source
+        # dest, source = source, dest # alternate dest and source
+        source = dest
     end
     _unsafe_mul!(y, last(A.maps), source)
     return y
diff --git a/src/scaledmap.jl b/src/scaledmap.jl
index 549bf7a8..d6f82eb9 100644
--- a/src/scaledmap.jl
+++ b/src/scaledmap.jl
@@ -7,14 +7,16 @@ struct ScaledMap{T, S<:RealOrComplex, L<:LinearMap} <: LinearMap{T}
     λ::S
     lmap::L
     function ScaledMap{T}(λ::S, A::L) where {T, S <: RealOrComplex, L <: LinearMap{<:RealOrComplex}}
-        @assert Base.promote_op(*, S, eltype(A)) == T "target type $T cannot hold products of $S and $(eltype(A)) objects"
+        Tprod = typeof(oneunit(S) * oneunit(eltype(A)))
+        promote_type(T, Tprod) == T ||
+           error("target type $T vs product of $S and $(eltype(A))")
         new{T,S,L}(λ, A)
     end
 end
 
 # constructor
 ScaledMap(λ::RealOrComplex, lmap::LinearMap{<:RealOrComplex}) =
-    ScaledMap{Base.promote_op(*, typeof(λ), eltype(lmap))}(λ, lmap)
+    ScaledMap{typeof(oneunit(λ) * oneunit(eltype(lmap)))}(λ, lmap)
 
 # basic methods
 Base.size(A::ScaledMap) = size(A.lmap)
diff --git a/test/Project.toml b/test/Project.toml
index c3704248..9155a6df 100644
--- a/test/Project.toml
+++ b/test/Project.toml
@@ -11,6 +11,7 @@ Quaternions = "94ee1d12-ae83-5a48-8b1c-48b8ff168ae0"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+Unitful = "1986cc42-f94f-5a68-af5c-568840ba703d"
 
 [compat]
 Aqua = "0.5, 0.6"
diff --git a/test/runtests.jl b/test/runtests.jl
index bde3fd94..fc473236 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -42,6 +42,8 @@ include("inversemap.jl")
 
 include("rrules.jl")
 
+include("units.jl")
+
 include("khatrirao.jl")
 
 include("trace.jl")
diff --git a/test/scaledmap.jl b/test/scaledmap.jl
index 8a35f27e..bd06e157 100644
--- a/test/scaledmap.jl
+++ b/test/scaledmap.jl
@@ -41,7 +41,7 @@ using Test, LinearMaps, LinearAlgebra
     # complex case
     β = π + 2π * im
     C = @inferred β * A
-    @test_throws AssertionError LinearMaps.ScaledMap{Float64}(β, A)
+    @test_throws ErrorException LinearMaps.ScaledMap{Float64}(β, A)
     @inferred conj(β) * A' # needed in left-mul
     T = ComplexF64
     xc = rand(T, N)
diff --git a/test/units.jl b/test/units.jl
new file mode 100644
index 00000000..f2c8119f
--- /dev/null
+++ b/test/units.jl
@@ -0,0 +1,26 @@
+# test/units
+
+using Test: @test, @testset, @inferred, @test_throws
+using LinearMaps: LinearMap
+using Unitful: m, s, g
+
+@testset "units" begin
+    A = rand(4,3) * 1m
+    B = rand(3,2) * 1s
+    C = A * B
+    D = 1f0g * C
+
+    Ma = @inferred LinearMap(A)
+    Mb = @inferred LinearMap(B)
+    Mc = Ma * Mb
+    Md = 1f0g * Mc
+
+    @test Matrix(Ma) == A
+    @test Matrix(Mc) == C
+    @test Matrix(Md) == D
+
+    x = randn(2)
+    @test B * x == Mb * x
+    @test C * x ≈ Mc * x
+    @test D * x ≈ Md * x
+end