Many computer vision problems require optimization of binary non-submodular energies. We propose a general optimization framework based on local submodular approximations (LSA). Unlike standard LP relaxation methods that linearize the whole energy globally, our approach iteratively approximates the energies locally. On the other hand, unlike standard local optimization methods (e.g. gradient descent or projection techniques) we use non-linear submodular approximations and optimize them without leaving the domain of integer solutions. We discuss two specific LSA algorithms based on trust region and auxiliary function principles, LSA-TR and LSA-AUX. These methods obtain state-of-the-art results on a wide range of applications outperforming many standard techniques such as LBP, QPBO, and TRWS. While our paper is focused on pairwise energies, our ideas extend to higher-order problems. The code is available online (http://vision.csd.uwo.ca/code/).