The moduli space of a certain class of geometric objects parameterizes the isomorphism classes of such objects. Moduli spaces occur naturally in classification problems. They appear in many branches of mathematics and in particular in algebraic geometry. For a natural number $g > 1$ the moduli space $M_g$ classifies smooth projective curves of genus $g$. In 1961 Deligne and Mumford introduced a compactification of this space by means of stable curves. The geometry of these moduli spaces have been studied extensively since then by people from different perspectives. In this mini course we will give a review of well-known facts and conjectures about moduli spaces of curves and their invariants. Our main focus is on the study of the so called tautological classes on these moduli spaces in Chow and cohomology.