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3D triangle meshes are a common form for representing the geometry of static and dynamic 3D objects. They are employed already in many areas, e.g. e-commerce, video games, online museums, CGI or 3D animated films, etc. Static triangle meshes represent only a piecewise linear approximation of complex 3D objects. As a consequence the approximation error can be unacceptably high unless the number of triangles is sufficiently large. On the other hand a large number of triangles makes these meshes cumbersome to handle and expensive to store or to transmit. Consequently, there exists a demand for techniques for efficient compression of static and dynamic 3D meshes. In this article we start with basics on 3D meshes. Thereafter, we explain the key ideas behind different mesh compression approaches for static and dynamic 3D meshes, and highlight their similarities and differences. Finally, we introduce the upcoming MPEG standard for compression of dynamic 3D meshes, which is referred to as FAMC (Frame-based Animated Mesh Compression), and show comparative compression results.