$M_{g,n}$ is an algebraic variety which parameterises isomorphism classes of smooth curves of genus g and n marked points. The study of $M_{g,n}$ goes back to Riemann in 1857 and has been an object of study ever since, although the first rigorous construction is due to Mumford in 1965. We give a gentle introduction to the construction of $M_{g,n}$ and the techniques used to study its geometry.