We develop an algorithm for meshing molecular surfaces that is based on patch-wise meshing using an advancing-front method adapted to the particular case of molecular surfaces. We focus on the solvent accessible surface (SAS) and the solvent excluded surface (SES). The essential ingredient is a newly developed analysis of such surfaces in [18] that allows to describe all SES-singularities a priori and therefore a complete characterization of the SES. In addition, an algorithm for filling molecular inner holes is proposed based on the pre-computed data structures of molecular surfaces.