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.