Deep learning has met key applications in image computing, but still lacks processing paradigms for meshes, i.e. collections of elementary geometrical parts such as points, segments or triangles. Meshes are both a powerful representation for geometrical objects, and a challenge for network architectures because of their inherent irregular structure. This work contributes to adapt classical deep learning paradigms to this particular type of data in three ways. First, we introduce the current-splatting layer which embeds meshes in a metric space, allowing the downstream network to process them without any assumption on their topology: they may be composed of varied numbers of elements or connected components, contain holes, or bear high levels of geometrical noise. Second, we adapt to meshes the exponentiation layer which, from an upstream image array, generates shapes with a diffeomorphic control over their topology. Third, we take advantage of those layers to devise a variational auto-encoding architecture, which we interpret as a generative statistical model that learns adapted low-dimensional representations for mesh data sets. An explicit norm-control layer ensures the correspondence between the latent-space Euclidean metric and the shape-space log-Euclidean one. We illustrate this method on simulated and real data sets, and show the practical relevance of the learned representation for visualization, classification and mesh synthesis.