Many reaction-diffusion systems in biology and chemistry give rise to patterns that are localized in space. Examples of these include models of chemo-aggregation, of phase transitions in materials science, models of spot and stripe formation in morphogenesis, and models of excitable media involving spiral waves. The analysis of the stability, existence, and dynamics of these localized structures requires much more mathematical machinery than the conventional Turing stability analysis of spatially uniform steady states. I will discuss a few of these topics and survey a few recent results. I will also list a few open problems.