Shack-Hartmann wavefront sensors (SHWFSs) are a popular tool that can be used to derive the point spread function of an optical system, but the speed of SHWFSs are limited by both the noise in the system and the underlying camera hardware. The marriage of sparse reconstruction algorithms with wavefront sensing has the capacity to improve the speed of SHWFSs by changing only the way that data are acquired and processed. By reducing the number of lenslets required to construct an optical wavefront, fewer pixels on the SHWFS camera need to be read and recorded. Reading fewer pixels increases the frame rate at which the sensor can operate without altering the physical hardware of current SHWFS devices. We have developed an algorithm for fast wavefront sensing that incorporates sparse representation for the first time in practice. Optical wavefronts are reconstructed with as little as 5% of the original data using a sparse Zernike representation (SPARZER). Compressed wavefront sensing offers the potential to increase the speed of wavefront acquisition and to defray the cost of SHWFS devices.