diff --git a/base/broadcast.jl b/base/broadcast.jl
index 24d0424a89727..adb9a1b662a75 100644
--- a/base/broadcast.jl
+++ b/base/broadcast.jl
@@ -166,7 +166,7 @@ BroadcastStyle(a::AbstractArrayStyle{M}, ::DefaultArrayStyle{N}) where {M,N} =
 # methods that instead specialize on `BroadcastStyle`,
 #    copyto!(dest::AbstractArray, bc::Broadcasted{MyStyle})
 
-struct Broadcasted{Style<:Union{Nothing,BroadcastStyle}, Axes, F, Args<:Tuple}
+struct Broadcasted{Style<:Union{Nothing,BroadcastStyle}, Axes, F, Args<:Tuple} <: Base.AbstractBroadcasted
     f::F
     args::Args
     axes::Axes          # the axes of the resulting object (may be bigger than implied by `args` if this is nested inside a larger `Broadcasted`)
@@ -219,6 +219,18 @@ argtype(bc::Broadcasted) = argtype(typeof(bc))
 _eachindex(t::Tuple{Any}) = t[1]
 _eachindex(t::Tuple) = CartesianIndices(t)
 
+_combined_indexstyle(::Type{Tuple{A}}) where A = IndexStyle(A)
+_combined_indexstyle(::Type{TT}) where TT =
+    IndexStyle(IndexStyle(Base.tuple_type_head(TT)),
+               _combined_indexstyle(Base.tuple_type_tail(TT)))
+
+Base.IndexStyle(bc::Broadcasted) = IndexStyle(typeof(bc))
+Base.IndexStyle(::Type{<:Broadcasted{<:Any,Nothing,<:Any,Args}}) where {Args <: Tuple} =
+    _combined_indexstyle(Args)
+Base.IndexStyle(::Type{<:Broadcasted{<:Any}}) = IndexCartesian()
+
+Base.LinearIndices(bc::Broadcasted) = LinearIndices(eachindex(bc))
+
 Base.ndims(::Broadcasted{<:Any,<:NTuple{N,Any}}) where {N} = N
 Base.ndims(::Type{<:Broadcasted{<:Any,<:NTuple{N,Any}}}) where {N} = N
 
diff --git a/base/reduce.jl b/base/reduce.jl
index 63694299d7da3..d3ad08dcf1db0 100644
--- a/base/reduce.jl
+++ b/base/reduce.jl
@@ -12,6 +12,9 @@ else
     const SmallUnsigned = Union{UInt8,UInt16,UInt32}
 end
 
+abstract type AbstractBroadcasted end
+const AbstractArrayOrBroadcasted = Union{AbstractArray, AbstractBroadcasted}
+
 """
     Base.add_sum(x, y)
 
@@ -152,7 +155,8 @@ foldr(op, itr; kw...) = mapfoldr(identity, op, itr; kw...)
 
 # This is a generic implementation of `mapreduce_impl()`,
 # certain `op` (e.g. `min` and `max`) may have their own specialized versions.
-@noinline function mapreduce_impl(f, op, A::AbstractArray, ifirst::Integer, ilast::Integer, blksize::Int)
+@noinline function mapreduce_impl(f, op, A::AbstractArrayOrBroadcasted,
+                                  ifirst::Integer, ilast::Integer, blksize::Int)
     if ifirst == ilast
         @inbounds a1 = A[ifirst]
         return mapreduce_first(f, op, a1)
@@ -175,7 +179,7 @@ foldr(op, itr; kw...) = mapfoldr(identity, op, itr; kw...)
     end
 end
 
-mapreduce_impl(f, op, A::AbstractArray, ifirst::Integer, ilast::Integer) =
+mapreduce_impl(f, op, A::AbstractArrayOrBroadcasted, ifirst::Integer, ilast::Integer) =
     mapreduce_impl(f, op, A, ifirst, ilast, pairwise_blocksize(f, op))
 
 """
@@ -296,13 +300,13 @@ The default is `reduce_first(op, f(x))`.
 """
 mapreduce_first(f, op, x) = reduce_first(op, f(x))
 
-_mapreduce(f, op, A::AbstractArray) = _mapreduce(f, op, IndexStyle(A), A)
+_mapreduce(f, op, A::AbstractArrayOrBroadcasted) = _mapreduce(f, op, IndexStyle(A), A)
 
-function _mapreduce(f, op, ::IndexLinear, A::AbstractArray{T}) where T
+function _mapreduce(f, op, ::IndexLinear, A::AbstractArrayOrBroadcasted)
     inds = LinearIndices(A)
     n = length(inds)
     if n == 0
-        return mapreduce_empty(f, op, T)
+        return mapreduce_empty_iter(f, op, A, IteratorEltype(A))
     elseif n == 1
         @inbounds a1 = A[first(inds)]
         return mapreduce_first(f, op, a1)
@@ -323,7 +327,7 @@ end
 
 mapreduce(f, op, a::Number) = mapreduce_first(f, op, a)
 
-_mapreduce(f, op, ::IndexCartesian, A::AbstractArray) = mapfoldl(f, op, A)
+_mapreduce(f, op, ::IndexCartesian, A::AbstractArrayOrBroadcasted) = mapfoldl(f, op, A)
 
 """
     reduce(op, itr; [init])
@@ -473,7 +477,7 @@ isgoodzero(::typeof(max), x) = isbadzero(min, x)
 isgoodzero(::typeof(min), x) = isbadzero(max, x)
 
 function mapreduce_impl(f, op::Union{typeof(max), typeof(min)},
-                        A::AbstractArray, first::Int, last::Int)
+                        A::AbstractArrayOrBroadcasted, first::Int, last::Int)
     a1 = @inbounds A[first]
     v1 = mapreduce_first(f, op, a1)
     v2 = v3 = v4 = v1
@@ -746,7 +750,7 @@ function count(pred, itr)
     end
     return n
 end
-function count(pred, a::AbstractArray)
+function count(pred, a::AbstractArrayOrBroadcasted)
     n = 0
     for i in eachindex(a)
         @inbounds n += pred(a[i])::Bool
diff --git a/base/reducedim.jl b/base/reducedim.jl
index 7b005aeff4f10..14d6a6bfb2069 100644
--- a/base/reducedim.jl
+++ b/base/reducedim.jl
@@ -301,16 +301,19 @@ julia> mapreduce(isodd, |, a, dims=1)
  1  1  1  1
 ```
 """
-mapreduce(f, op, A::AbstractArray; dims=:, kw...) = _mapreduce_dim(f, op, kw.data, A, dims)
+mapreduce(f, op, A::AbstractArrayOrBroadcasted; dims=:, kw...) =
+    _mapreduce_dim(f, op, kw.data, A, dims)
 
-_mapreduce_dim(f, op, nt::NamedTuple{(:init,)}, A::AbstractArray, ::Colon) = mapfoldl(f, op, A; nt...)
+_mapreduce_dim(f, op, nt::NamedTuple{(:init,)}, A::AbstractArrayOrBroadcasted, ::Colon) =
+    mapfoldl(f, op, A; nt...)
 
-_mapreduce_dim(f, op, ::NamedTuple{()}, A::AbstractArray, ::Colon) = _mapreduce(f, op, IndexStyle(A), A)
+_mapreduce_dim(f, op, ::NamedTuple{()}, A::AbstractArrayOrBroadcasted, ::Colon) =
+    _mapreduce(f, op, IndexStyle(A), A)
 
-_mapreduce_dim(f, op, nt::NamedTuple{(:init,)}, A::AbstractArray, dims) =
+_mapreduce_dim(f, op, nt::NamedTuple{(:init,)}, A::AbstractArrayOrBroadcasted, dims) =
     mapreducedim!(f, op, reducedim_initarray(A, dims, nt.init), A)
 
-_mapreduce_dim(f, op, ::NamedTuple{()}, A::AbstractArray, dims) =
+_mapreduce_dim(f, op, ::NamedTuple{()}, A::AbstractArrayOrBroadcasted, dims) =
     mapreducedim!(f, op, reducedim_init(f, op, A, dims), A)
 
 """