Grothendieck showed that morphisms of projective varieties are parameterized by a scheme. This fails tremendously for affine varieties. However, group structures tend to be rigidifying, and indeed homomorphisms of affine groups are sometimes also parameterized by a scheme. In this talk, I will discuss the general problem of parameterizing homomorphisms of algebraic groups, both over a field (at first) and over a general base (by the end). Over a field, the main result has a surprisingly elementary proof, which we will describe. We will also give several examples throughout which illustrate strange phenomena that crop up.