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In this article we give a survey of homology computations for moduli spaces M m g,1 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.