The goal of these lectures is to introduce and motivate the construction of the moduli spaces of curves, as well as strictly related basic important concepts (such as orbifolds and stacks). The first part of this mini-course will focus on the complex analytic aspects of the study of moduli spaces of curves (Teichmüller spaces, Beltrami equation, quadratic differentials). The second part of the mini-course will mainly focus on the algebro-geometric aspects (Grothendieck’s construction of Teichmüller spaces, representability of functors, a gentle introduction to stacks). If time allows, more recent results on the intersection theory on moduli spaces of curves will be presented (e.g. cohomological field theories, Witten’s conjecture & Kontsevich Theorem, Faber’s conjecture, relations in the tautological ring).