Stabilizer states are a rich class of quantum states that can be efficiently represented and manipulated on classical computers. This feature makes stabilizer states a useful basis for simulating quantum computations that do not deviate too far from a sequence of Clifford operations, for instance noisy quantum error correction circuits, noise-free circuits with few T gates, and some low-depth variational circuits. In this talk I will review several approaches to stabilizer-based simulation and some of the tradeoffs involved. I will present a hybrid method, combining several previously known techniques, for efficiently simulating near-Clifford circuits with near-Clifford noise. Finally, I will present results showing the application of this method to variational quantum computations.