The aim of the talk is to introduce algebraic stacks and use the example of the moduli space of curves as motivation. Firstly I will talk about moduli problems in geometry and the corresponding categorical notion of representable functor. As the moduli functor of curves is not representable, there are two possible ways to bypass the problem: either consider coarse moduli spaces or enlarge the category of geometric objects considered and introduce ‘stacks’, which are more or less the algebraic counterpart of orbifolds in differential topology. Finally I will state some geometric properties of the stack M_g (the stack of smooth curves of genus g) and I will show how they can be formulated in terms of geometric properties of families of curves.