An overview of the physics of spinor and dipolar Bose-Einstein condensates (BECs) is presented. Mean-field ground states, Bogoliubov spectra, and many-body ground and excited states of spinor BECs are discussed. Properties of spin-polarized dipolar BECs and those of spinor-dipolar BECs are reviewed. Some of the unique features of the vortices in spinor BECs, such as fractional vortices and non-Abelian vortices, are delineated. The symmetry of the order parameter is classified using group theory, and various topological excitations are investigated based on homotopy theory. Some of the more recent developments such as the Kibble-Zurek mechanism of defect formation and the Berezinskii-Kosterlitz-Thouless transition in a spinor BEC are also discussed.