  We give two results concerning the construction of modular invariant
partition functions for conformal field theories constructed by tensoring
together other conformal field theories. First we show how the possible modular
invariants for the tensor product theory are constrained if the allowed modular
invariants of the individual conformal field theory factors have been
classified. We illustrate the use of these constraints for theories of the type
$SU(2)_{K_A}\otimes SU(2)_{K_B}$, finding all consistent theories for $K_A,K_B$
odd. Second we show how known diagonal modular invariants can be used to
construct some inherently asymmetric ones where the holomorphic and
anti-holomorphic theories do not share the same chiral algebra. Some explicit
examples are given.
