The Laplacian matrices of graphs arise in many fields including Machine Learning, Computer Vision, Optimization, Computational Science, and of course Network Analysis. We will explain what these matrices are and why they arise in so many applications. We then will survey recent progress on the design of algorithms that allow us to solve such systems of linear equations in nearly linear time. In particular, we will show how fast algorithms for graph sparsification directly lead to fast Laplacian system solvers. As an application, we will explain how Laplacian system solvers can be used to quickly solve linear programs arising from natural graph problems.