It is well-known that the triangulations of the disc with n + 2 vertices on its boundary are counted by the nth Catalan number C(n) = 1 n+1 (2n n ) . This paper deals with the generalisation of this problem to any compact surface S with boundaries. We obtain the asymptotic number of simplicial decompositions of the surface S with n vertices on its boundary. More generally, we determine the asymptotic number of dissections of S when the faces are d-gons with d belonging to a set of admissible degrees ¿ ¿ {3, 4, 5, . . .}. We also give the limit laws for certain parameters of such dissections