Super-resolution localization microscopy methods provide powerful new capabilities for probing biology at the nanometer scale via fluorescence. These methods rely on two key innovations: switchable fluorophores (which blink on and off and can be sequentially imaged) and powerful localization algorithms (which estimate the positions of the fluorophores in the images). These techniques have spurred a flurry of innovation in algorithm development over the last several years. In this Review, we survey the fundamental issues for single-fluorophore fitting routines, localization algorithms based on principles other than fitting, three-dimensional imaging, dipole imaging and techniques for estimating fluorophore positions from images of multiple activated fluorophores. We offer practical advice for users and adopters of algorithms, and we identify areas for further development.