This paper follows a previous one that was dealing with high-quality surface remeshing using harmonic maps (Int. J. Numer. Meth. Engng 2010; 83:403–425). In (Int. J. Numer. Meth. Engng 2010; 83:403–425), it has been demonstrated that harmonic parametrizations can be used as input for surface meshers to produce high-quality triangulations. However, two important limitations were pointed out, namely surfaces with high genus and/or of large aspect ratio. This paper addresses those two issues. We first develop a multiscale version of the harmonic parametrization of (Int. J. Numer. Meth. Engng 2010; 83:403–425) and then combine it with a multilevel partitioning algorithm to come up with an automatic remeshing algorithm that overcomes the above-mentioned limitations of harmonic maps. The overall procedure is implemented in the open-source mesh generator Gmsh (Int. J. Numer. Meth. Engng 2009; 79(11): 1309–1331). Copyright © 2011 John Wiley & Sons, Ltd.