diff --git a/src/triangular.jl b/src/triangular.jl
index 26ad4204..64bf85e4 100644
--- a/src/triangular.jl
+++ b/src/triangular.jl
@@ -238,10 +238,22 @@ Base.isassigned(A::UpperOrLowerTriangular, i::Int, j::Int) =
 Base.isstored(A::UpperOrLowerTriangular, i::Int, j::Int) =
     _shouldforwardindex(A, i, j) ? Base.isstored(A.data, i, j) : false
 
-@propagate_inbounds getindex(A::Union{UnitLowerTriangular{T}, UnitUpperTriangular{T}}, i::Int, j::Int) where {T} =
-    _shouldforwardindex(A, i, j) ? A.data[i,j] : ifelse(i == j, oneunit(T), zero(T))
-@propagate_inbounds getindex(A::Union{LowerTriangular, UpperTriangular}, i::Int, j::Int) =
-    _shouldforwardindex(A, i, j) ? A.data[i,j] : diagzero(A,i,j)
+@propagate_inbounds function getindex(A::Union{UnitLowerTriangular{T}, UnitUpperTriangular{T}}, i::Int, j::Int) where {T}
+    if _shouldforwardindex(A, i, j)
+        A.data[i,j]
+    else
+        @boundscheck checkbounds(A, i, j)
+        ifelse(i == j, oneunit(T), zero(T))
+    end
+end
+@propagate_inbounds function getindex(A::Union{LowerTriangular, UpperTriangular}, i::Int, j::Int)
+    if _shouldforwardindex(A, i, j)
+        A.data[i,j]
+    else
+        @boundscheck checkbounds(A, i, j)
+        @inbounds diagzero(A,i,j)
+    end
+end
 
 _shouldforwardindex(U::UpperTriangular, b::BandIndex) = b.band >= 0
 _shouldforwardindex(U::LowerTriangular, b::BandIndex) = b.band <= 0
@@ -250,63 +262,97 @@ _shouldforwardindex(U::UnitLowerTriangular, b::BandIndex) = b.band < 0
 
 # these specialized getindex methods enable constant-propagation of the band
 Base.@constprop :aggressive @propagate_inbounds function getindex(A::Union{UnitLowerTriangular{T}, UnitUpperTriangular{T}}, b::BandIndex) where {T}
-    _shouldforwardindex(A, b) ? A.data[b] : ifelse(b.band == 0, oneunit(T), zero(T))
+    if _shouldforwardindex(A, b)
+        A.data[b]
+    else
+        @boundscheck checkbounds(A, b)
+        ifelse(b.band == 0, oneunit(T), zero(T))
+    end
 end
 Base.@constprop :aggressive @propagate_inbounds function getindex(A::Union{LowerTriangular, UpperTriangular}, b::BandIndex)
-    _shouldforwardindex(A, b) ? A.data[b] : diagzero(A.data, b)
+    if _shouldforwardindex(A, b)
+        A.data[b]
+    else
+        @boundscheck checkbounds(A, b)
+        @inbounds diagzero(A, b)
+    end
 end
 
-_zero_triangular_half_str(::Type{<:UpperOrUnitUpperTriangular}) = "lower"
-_zero_triangular_half_str(::Type{<:LowerOrUnitLowerTriangular}) = "upper"
-
-@noinline function throw_nonzeroerror(T, @nospecialize(x), i, j)
-    Ts = _zero_triangular_half_str(T)
-    Tn = nameof(T)
+@noinline function throw_nonzeroerror(Tn::Symbol, @nospecialize(x), i, j)
+    zero_half = Tn in (:UpperTriangular, :UnitUpperTriangular) ? "lower" : "upper"
+    nstr = Tn === :UpperTriangular ? "n" : ""
     throw(ArgumentError(
-        lazy"cannot set index in the $Ts triangular part ($i, $j) of an $Tn matrix to a nonzero value ($x)"))
+        LazyString(
+            lazy"cannot set index ($i, $j) in the $zero_half triangular part ",
+            lazy"of a$nstr $Tn matrix to a nonzero value ($x)")
+        )
+    )
 end
-@noinline function throw_nononeerror(T, @nospecialize(x), i, j)
-    Tn = nameof(T)
+@noinline function throw_nonuniterror(Tn::Symbol, @nospecialize(x), i, j)
     throw(ArgumentError(
-        lazy"cannot set index on the diagonal ($i, $j) of an $Tn matrix to a non-unit value ($x)"))
+        lazy"cannot set index ($i, $j) on the diagonal of a $Tn matrix to a non-unit value ($x)"))
 end
 
 @propagate_inbounds function setindex!(A::UpperTriangular, x, i::Integer, j::Integer)
-    if i > j
-        iszero(x) || throw_nonzeroerror(typeof(A), x, i, j)
-    else
+    if _shouldforwardindex(A, i, j)
         A.data[i,j] = x
+    else
+        @boundscheck checkbounds(A, i, j)
+        # the value must be convertible to the eltype for setindex! to be meaningful
+        # however, the converted value is unused, and the compiler is free to remove
+        # the conversion if the call is guaranteed to succeed
+        convert(eltype(A), x)
+        iszero(x) || throw_nonzeroerror(nameof(typeof(A)), x, i, j)
     end
     return A
 end
 
 @propagate_inbounds function setindex!(A::UnitUpperTriangular, x, i::Integer, j::Integer)
-    if i > j
-        iszero(x) || throw_nonzeroerror(typeof(A), x, i, j)
-    elseif i == j
-        x == oneunit(x) || throw_nononeerror(typeof(A), x, i, j)
-    else
+    if _shouldforwardindex(A, i, j)
         A.data[i,j] = x
+    else
+        @boundscheck checkbounds(A, i, j)
+        # the value must be convertible to the eltype for setindex! to be meaningful
+        # however, the converted value is unused, and the compiler is free to remove
+        # the conversion if the call is guaranteed to succeed
+        convert(eltype(A), x)
+        if i == j # diagonal
+            x == oneunit(eltype(A)) || throw_nonuniterror(nameof(typeof(A)), x, i, j)
+        else
+            iszero(x) || throw_nonzeroerror(nameof(typeof(A)), x, i, j)
+        end
     end
     return A
 end
 
 @propagate_inbounds function setindex!(A::LowerTriangular, x, i::Integer, j::Integer)
-    if i < j
-        iszero(x) || throw_nonzeroerror(typeof(A), x, i, j)
-    else
+    if _shouldforwardindex(A, i, j)
         A.data[i,j] = x
+    else
+        @boundscheck checkbounds(A, i, j)
+        # the value must be convertible to the eltype for setindex! to be meaningful
+        # however, the converted value is unused, and the compiler is free to remove
+        # the conversion if the call is guaranteed to succeed
+        convert(eltype(A), x)
+        iszero(x) || throw_nonzeroerror(nameof(typeof(A)), x, i, j)
     end
     return A
 end
 
 @propagate_inbounds function setindex!(A::UnitLowerTriangular, x, i::Integer, j::Integer)
-    if i < j
-        iszero(x) || throw_nonzeroerror(typeof(A), x, i, j)
-    elseif i == j
-        x == oneunit(x) || throw_nononeerror(typeof(A), x, i, j)
-    else
+    if _shouldforwardindex(A, i, j)
         A.data[i,j] = x
+    else
+        @boundscheck checkbounds(A, i, j)
+        # the value must be convertible to the eltype for setindex! to be meaningful
+        # however, the converted value is unused, and the compiler is free to remove
+        # the conversion if the call is guaranteed to succeed
+        convert(eltype(A), x)
+        if i == j  # diagonal
+            x == oneunit(eltype(A)) || throw_nonuniterror(nameof(typeof(A)), x, i, j)
+        else
+            iszero(x) || throw_nonzeroerror(nameof(typeof(A)), x, i, j)
+        end
     end
     return A
 end
@@ -560,7 +606,7 @@ for (T, UT) in ((:UpperTriangular, :UnitUpperTriangular), (:LowerTriangular, :Un
     @eval @inline function _copy!(A::$UT, B::$T)
         for dind in diagind(A, IndexStyle(A))
             if A[dind] != B[dind]
-                throw_nononeerror(typeof(A), B[dind], Tuple(dind)...)
+                throw_nonuniterror(nameof(typeof(A)), B[dind], Tuple(dind)...)
             end
         end
         _copy!($T(parent(A)), B)
@@ -741,7 +787,7 @@ function _triscale!(A::UpperOrUnitUpperTriangular, B::UnitUpperTriangular, c::Nu
     checksize1(A, B)
     _iszero_alpha(_add) && return _rmul_or_fill!(A, _add.beta)
     for j in axes(B.data,2)
-        @inbounds _modify!(_add, c, A, (j,j))
+        @inbounds _modify!(_add, B[BandIndex(0,j)] * c, A, (j,j))
         for i in firstindex(B.data,1):(j - 1)
             @inbounds _modify!(_add, B.data[i,j] * c, A.data, (i,j))
         end
@@ -752,7 +798,7 @@ function _triscale!(A::UpperOrUnitUpperTriangular, c::Number, B::UnitUpperTriang
     checksize1(A, B)
     _iszero_alpha(_add) && return _rmul_or_fill!(A, _add.beta)
     for j in axes(B.data,2)
-        @inbounds _modify!(_add, c, A, (j,j))
+        @inbounds _modify!(_add, c * B[BandIndex(0,j)], A, (j,j))
         for i in firstindex(B.data,1):(j - 1)
             @inbounds _modify!(_add, c * B.data[i,j], A.data, (i,j))
         end
@@ -783,7 +829,7 @@ function _triscale!(A::LowerOrUnitLowerTriangular, B::UnitLowerTriangular, c::Nu
     checksize1(A, B)
     _iszero_alpha(_add) && return _rmul_or_fill!(A, _add.beta)
     for j in axes(B.data,2)
-        @inbounds _modify!(_add, c, A, (j,j))
+        @inbounds _modify!(_add, B[BandIndex(0,j)] * c, A, (j,j))
         for i in (j + 1):lastindex(B.data,1)
             @inbounds _modify!(_add, B.data[i,j] * c, A.data, (i,j))
         end
@@ -794,7 +840,7 @@ function _triscale!(A::LowerOrUnitLowerTriangular, c::Number, B::UnitLowerTriang
     checksize1(A, B)
     _iszero_alpha(_add) && return _rmul_or_fill!(A, _add.beta)
     for j in axes(B.data,2)
-        @inbounds _modify!(_add, c, A, (j,j))
+        @inbounds _modify!(_add, c * B[BandIndex(0,j)], A, (j,j))
         for i in (j + 1):lastindex(B.data,1)
             @inbounds _modify!(_add, c * B.data[i,j], A.data, (i,j))
         end
diff --git a/test/triangular.jl b/test/triangular.jl
index 21fc2d69..0ad4f521 100644
--- a/test/triangular.jl
+++ b/test/triangular.jl
@@ -641,11 +641,11 @@ end
         @testset "error message" begin
             A = UpperTriangular(Ap)
             B = UpperTriangular(Bp)
-            @test_throws "cannot set index in the lower triangular part" copyto!(A, B)
+            @test_throws "cannot set index (3, 1) in the lower triangular part" copyto!(A, B)
 
             A = LowerTriangular(Ap)
             B = LowerTriangular(Bp)
-            @test_throws "cannot set index in the upper triangular part" copyto!(A, B)
+            @test_throws "cannot set index (1, 2) in the upper triangular part" copyto!(A, B)
         end
     end
 
@@ -950,6 +950,10 @@ end
     @test 2\U == 2\M
     @test U*2 == M*2
     @test 2*U == 2*M
+
+    U2 = copy(U)
+    @test rmul!(U, 1) == U2
+    @test lmul!(1, U) == U2
 end
 
 @testset "scaling partly initialized unit triangular" begin
@@ -966,4 +970,75 @@ end
     end
 end
 
+@testset "indexing checks" begin
+    P = [1 2; 3 4]
+    @testset "getindex" begin
+        U = UnitUpperTriangular(P)
+        @test_throws BoundsError U[0,0]
+        @test_throws BoundsError U[1,0]
+        @test_throws BoundsError U[BandIndex(0,0)]
+        @test_throws BoundsError U[BandIndex(-1,0)]
+
+        U = UpperTriangular(P)
+        @test_throws BoundsError U[1,0]
+        @test_throws BoundsError U[BandIndex(-1,0)]
+
+        L = UnitLowerTriangular(P)
+        @test_throws BoundsError L[0,0]
+        @test_throws BoundsError L[0,1]
+        @test_throws BoundsError U[BandIndex(0,0)]
+        @test_throws BoundsError U[BandIndex(1,0)]
+
+        L = LowerTriangular(P)
+        @test_throws BoundsError L[0,1]
+        @test_throws BoundsError L[BandIndex(1,0)]
+    end
+    @testset "setindex!" begin
+        A = SizedArrays.SizedArray{(2,2)}(P)
+        M = fill(A, 2, 2)
+        U = UnitUpperTriangular(M)
+        @test_throws "Cannot `convert` an object of type $Int" U[1,1] = 1
+        non_unit_msg = "cannot set index $((1,1)) on the diagonal of a UnitUpperTriangular matrix to a non-unit value"
+        @test_throws non_unit_msg U[1,1] = A
+        L = UnitLowerTriangular(M)
+        @test_throws "Cannot `convert` an object of type $Int" L[1,1] = 1
+        non_unit_msg = "cannot set index $((1,1)) on the diagonal of a UnitLowerTriangular matrix to a non-unit value"
+        @test_throws non_unit_msg L[1,1] = A
+
+        for UT in (UnitUpperTriangular, UpperTriangular)
+            U = UT(M)
+            @test_throws "Cannot `convert` an object of type $Int" U[2,1] = 0
+        end
+        for LT in (UnitLowerTriangular, LowerTriangular)
+            L = LT(M)
+            @test_throws "Cannot `convert` an object of type $Int" L[1,2] = 0
+        end
+
+        U = UnitUpperTriangular(P)
+        @test_throws BoundsError U[0,0] = 1
+        @test_throws BoundsError U[1,0] = 0
+
+        U = UpperTriangular(P)
+        @test_throws BoundsError U[1,0] = 0
+
+        L = UnitLowerTriangular(P)
+        @test_throws BoundsError L[0,0] = 1
+        @test_throws BoundsError L[0,1] = 0
+
+        L = LowerTriangular(P)
+        @test_throws BoundsError L[0,1] = 0
+    end
+end
+
+@testset "unit triangular l/rdiv!" begin
+    A = rand(3,3)
+    @testset for (UT,T) in ((UnitUpperTriangular, UpperTriangular),
+                            (UnitLowerTriangular, LowerTriangular))
+        UnitTri = UT(A)
+        Tri = T(LinearAlgebra.full(UnitTri))
+        @test 2 \ UnitTri ≈ 2 \ Tri
+        @test UnitTri / 2 ≈ Tri / 2
+    end
+end
+
 end # module TestTriangular