Submodular function minimization is a fundamental optimization problem that arises in several applications in machine learning and computer vision. The problem is known to be solvable in polynomial time, but general purpose algorithms have high running times and are unsuitable for large-scale problems. Recent work aims to obtain very practical algorithms for minimizing functions that are sums of "simple" functions. This class of functions provides an important bridge between functions with a rich combinatorial structure -- notably, the cut function of a graph -- and general submodular functions, and it brings along powerful combinatorial structure reminiscent of graphs as well as a fair bit of modeling power that greatly expands its applications. In this talk, we describe recent progress on designing very efficient algorithms for minimizing decomposable functions and understanding their combinatorial structure.