Exciton-polariton Bose-Einstein condensates (BECs) are newly emerged quantum systems that are capable of showing macroscopic quantum phenomena with intrinsic open-dissipative nature. The spatial distribution of the polariton density, without any external potential, can be controlled by the shape of the pumping laser beam, enabling the investigation of polariton dynamics with topologically non-trivial configurations. In addition, exciton-polaritons have a spin degree of freedom inherited from photons. In this talk, we will present theoretical results on nonlinear dynamics of an exciton-polariton BEC involving both polariton's spatial degrees of freedom and spin degrees of freedom, and interactions between them. We explore formation and dynamics of persistent currents and spin textures, and suggest techniques for their experimental observation in exciton-polariton condensates. Finally, the polariton Talbot effect resulting from trapping polariton condensates in an 1D mesa array will also be discussed.