Submodular functions are relevant to machine learning for mainly two reasons: (1) some problems may be expressed directly as the optimization of submodular functions and (2) the Lovasz extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In this talk, I will present the theory of submodular functions from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. In particular, I will show how submodular function minimization is equivalent to solving a wide variety of convex optimization problems. This allows the derivation of new efficient algorithms for approximate submodular function minimization with theoretical guarantees and good practical performance. By listing examples of submodular functions, I will also review various applications to machine learning, such as clustering or subset selection, as well as a family of structured sparsity-inducing norms that can be derived and used from submodular functions.