The idea of moduli space is quite intuitive, you have surely encountered it multiple times. We will present a general overview of moduli spaces in an algebraic context, stressing the categorical point of view. In the second part of the talk, we will focus on the Deligne-Mumford compactification of the moduli space of smooth curves that appeared in '69. Time permitting, we will discuss a generalization of coarse moduli spaces, due to Alper ('13), and present some examples.