chapter 1. Introduction -- chapter 2. Condensation and exciton-polaritons -- chapter 3. Experimental techniques and sample -- chapter 4. Synchronization of polariton condensates -- chapter 5. Condensation in multiple modes -- chapter 6. Conventional vorticity -- chapter 7. Unconventional vorticity -- chapter 8. Dynamics of spontaneous vortices -- chapter 9. Dynamics of polariton condensates in double wells -- chapter 10. Conclusions and outlook.