In this paper, we present an efficient and robust technique for surface remeshing based on harmonic maps. We show how to ensure a one-to-one mapping for the discrete harmonic map and introduce a cubic representation of the geometry based on curved PN triangles. Topological and geometrical limitations of harmonic maps are also put to the fore and discussed. We show that, with the proposed approach, we are able to recover high quality meshes from both low input STL triangulations and complex surfaces defined by many CAD patches. The overall procedure is implemented in the open-source mesh generator Gmsh. Copyright © 2010 John Wiley & Sons, Ltd.