In the two-dimensional O(n) and Potts models, some observables can be computed as weighted sums over configurations of non-intersecting loops. I will define weighted sums associated to a large class of combinatorial maps, also known as ribbon graphs, fatgraphs or rotation systems. Given a map with $N$ vertices, this yields a function of the moduli of the corresponding punctured Riemann surface, which I will call an $N$-point correlation function. I will conjecture that in the critical limit, such correlation functions form a basis of solutions of certain conformal bootstrap equations. They include all correlation functions of the O(n) and Potts models, and correlation functions that do not belong to any known model.