Algebraic geometry began its life as algebraic geometry over the real or complex numbers. In the 1940s and 50s mathematicians began to realize that it was possible, and fruitful, to think about algebraic geometry over other fields: number fields, for instance, or even characteristic p fields. Right now the field of algebraic topology finds itself in a similar situation. Rather than just studying classical topological spaces (which sort of amounts to studying topology over the reals of complexes) we have recently been learning to study topological "spaces" over other fields. The definitions are cumbersome and intimidating, much as the definition of schemes originally was, but the field is slowly yielding interesting results. I will attempt to give some kind of overview of this.