The moduli space of algebraic curves is a central object in algebraic geometry. The idea behind this space is that it answers to a classification problem, by allowing us to classify algebraic curves up to isomorphisms. Nonetheless, the geometry of this space is rather abstract and subtle, and only few general statements are known. 

The aim of this talk is to give an elementary introduction to the moduli space of npointed smooth curves of genus g and its compactification: we will define the main ingredients and present some basic properties. We will also discuss basic examples in order to get familiar with these spaces. 

One way to understand these spaces is by computing some topological invariants, such as their cohomology groups. In general, their full cohomology ring is still unknown, but we have a complete description for some values of g, which I will present in the final part.