Self-assembled vesicles are essential components of primitive cells. We review the importance of vesicles during the origins of life, fundamental thermodynamics and kinetics of self-assembly, and experimental models of simple vesicles, focusing on prebiotically plausible fatty acids and their derivatives. We review recent work on interactions of simple vesicles with RNA and other studies of the transition from vesicles to protocells. Finally we discuss current challenges in understanding the biophysics of protocells, as well as conceptual questions in information transmission and self-replication.