Let G be an algebraic group acting on an irreducible variety X. We say that a generic stabilizer has a given property if there is a nonempty open subset Y of X so that the stabilizer of every y in Y has the given property. In some situations, the generic stabilizer is a given conjugacy class of subgroups of G, but this is not always the case. We will discuss two situations in detail for G a simple algebraic group. The first is when X is a direct product of homogeneous spaces and the second is when X is a rational finite dimensional G-module. The situation in characteristic zero is well studied but we will focus on the case of positive characteristic.