Computations in the unstable homology of moduli spaces of Riemann surfaces

In this article we give a survey of homology computations for moduli spaces ${\mathfrak{M}}_{g,1}^m$ of Riemann surfaces with genus $g⩾0$, one boundary curve, and $m⩾0$ punctures. While rationally and stably this question has a satisfying answer by the Madsen-Weiss theorem, the unstable homology remains notoriously complicated. We discuss calculations with integral, mod-2, and rational coefficients. Furthermore, we determine, in most cases, explicit generators using homology operations.