Given λ, μ, and ν which are partitions of n, how many pairs (u, v) of permutations of [n] are there with u of cycle type μ, v of cycle type ν, and uv of cycle type λ? This problem has been called the "class multiplication problem" by Stanley. I will present recent work by Stanley and others on the case where μ = ν = {n} -- that is, on the cycle type of the product of two cycles. I will also speak about my own work in progress on products of involutions.