We consider the moduli space M_g,n of Riemann surfaces of genus g≥0 with n≥1 ordered and directed marked points. For d≥2g+n−1 we show that M_g,n is homotopy equivalent to a component of the simplicial Hurwitz space Hur^Δ(S_d^geo) associated with the partially multiplicative quandle S_d^geo. As an application, we give a new proof of the Mumford conjecture on the stable rational cohomology of moduli spaces of Riemann surfaces. We also provide a combinatorial model for the infinite loop space Ω^∞−2MTSO(2) of Hurwitz flavour.