the motivating problems in geometry are classification problems. one tries to understand the isomorphism classes of certain types of geometric objects. a peculiar feature of algebraic geometry is that there is often a variety or scheme, called the moduli space, whose points correspond to isomorphism classes of the objects we want to study. in this talk, i will talk about the history of moduli problems and spaces, and explain some of the mathematics behind the contributions grothendieck made to the subject.