Exciton-polaritons are hybrid light–matter quasiparticles in semiconductor microstructures that exhibit strong nonlinearity. An optical beam can efficiently drive polaritons to on-demand nonlinear states such as vortices and solitons. These states have rich physical properties, such as quantized phase jumps and stationary shapes, and hold promise for applications in information processing and transmission, all-optical circuits and quantum computing. In this Review, we discuss the formation and control dynamics of polariton vortices and solitons, with a focus on spinor polaritons in which vortices with a half-integer topological charge can arise.