A multiresolution triangle mesh is a geometric model providing representations of a surface at different levels of detail. A fairly general model is proposed, called a multiresolution triangulation (MT), capable of efficiently encoding a wide range of levels of detail. An MT supports efficient extraction of a representation of minimum size for a given application-dependent level of detail, which may be variable over the surface. The model is based on a collection of mesh fragments arranged into a partial order. Different levels of detail are obtained by combining different fragments according to rules controlled by the partial order. We illustrate general construction and extraction techniques, as well as applications of the MT in a variety of situations, ranging from the approximation of CAD surfaces to the reconstruction of unknown surfaces from scattered data.