"This talk discusses fast solvers for some large matrix problems which are rank structured. Examples of these structured problems include some large sparse discretized PDEs, Toeplitz systems, low-rank update eigenproblems, and others. Our fast algorithms use some semiseparable rank structured matrices. For these matrices, many matrix operations have linear complexity. I will first briefly show an example of a quadratic cost companion matrix eigensolver and condition estimator. Then I will focus on a fast multifrontal type direct solver for large sparse discretized linear systems. Semiseparable matrices are used to approximate dense intermediate matrices in the factorization. A new linear time factorization algorithm for semiseparable matrices is presented. The overall sparse solver has nearly linear complexity and linear storage, and has good potential for parallelization. It can also work as an effective preconditioner. Numerical results will be shown. This is joint work with Shiv Chandrasekaran, Ming Gu, Alan Laub, and Xiaoye Li."