Nowadays, digital 3D models are in widespread and ubiquitous use, and each specific application dealing with 3D geometry has its own quality requirements that restrict the class of acceptable and supported models. This article analyzes typical defects that make a 3D model unsuitable for key application contexts, and surveys existing algorithms that process, repair, and improve its structure, geometry, and topology to make it appropriate to case-by-case requirements. The analysis is focused on polygon meshes, which constitute by far the most common 3D object representation. In particular, this article provides a structured overview of mesh repairing techniques from the point of view of the application context. Different types of mesh defects are classified according to the upstream application that produced the mesh, whereas mesh quality requirements are grouped by representative sets of downstream applications where the mesh is to be used. The numerous mesh repair methods that have been proposed during the last two decades are analyzed and classified in terms of their capabilities, properties, and guarantees. Based on these classifications, guidelines can be derived to support the identification of repairing algorithms best-suited to bridge the compatibility gap between the quality provided by the upstream process and the quality required by the downstream applications in a given geometry processing scenario.