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Triangulated meshes have become ubiquitous discrete surface representations. In this paper, we address the problem of how to maintain the manifold properties of a surface while it undergoes strong deformations that may cause topological changes. We introduce a new self-intersection removal algorithm, TransforMesh, and propose a mesh evolution framework based on this algorithm. Numerous shape modeling applications use surface evolution in order to improve shape properties such as appearance or accuracy. Both explicit and implicit representations can be considered for that purpose. However, explicit mesh representations, while allowing for accurate surface modeling, suffer from the inherent difficulty of reliably dealing with self-intersections and topological changes such as merges and splits. As a consequence, a majority of methods rely on implicit representations of surfaces, e.g., level sets, that naturally overcome these issues. Nevertheless, these methods are based on volumetric discretizations, which introduce an unwanted precision-complexity trade-off. The method that we propose handles topological changes in a robust manner and removes self-intersections, thus overcoming the traditional limitations of mesh-based approaches. To illustrate the effectiveness of TransforMesh, we describe several challenging applications: surface morphing and 3D reconstruction.