Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement in many "easy" cases where other algorithms are doing too much work. Three ideas for adaptive polynomial multiplication are given. One method, which we call "chunky" multiplication, is given a more careful analysis, as well as an implementation in NTL. We show that significant improvements can be had over the fastest general-purpose algorithms in many cases.