Operator splitting schemes break a complicated and possible nonsmooth optimization problem into simple matrix-vector multiplication, gradient, projection, and proximal steps. The resulting algorithms are often short, easy to code, and have (nearly) state-of-the-art performance for large-scale optimization problems that arise in machine learning, compressed sensing, medical imaging, geophysics, and bioengineering. The importance of operator splitting, a technique that dates back to the 1950’ and since then been widely used in numerical linear algebra and numerical PDE, has significantly increased in the past decade. This talk will review the basic operator splitting schemes and introduce a new splitting scheme. Their special cases cover a large number of existing algorithms such as von Neumann's alternating projection, iterative soft-thresholding algorithm, ADMM, and various primal-dual algorithms. Their convergence results are presented. Through examples, we also demonstrate that they lead to high-performance low-cost methods for large-scale optimization problems. This talk includes joint work with Damek Davis, Wei Deng, Ming-jun Lai, Zhimin Peng, and Ming Yan.