We systematically investigate open strings in the plane wave background of type IIB string theory. We carefully analyze possible boundary conditions for open strings and find static as well as time-dependent branes. The branes fall into equivalence classes depending on whether they are related by the action of target space isometries. In particular static branes localized at the origin of transverse space and certain time-dependent branes fall into the same equivalence class. We analyze thoroughly the symmetries of all branes we discuss. Apart from symmetries descending from target space isometries, the worldsheet action being free admits a countably infinite number of other global worldsheet symmetries. We find that one can use such worldsheet symmetries to restore seemingly broken target space symmetries. In particular, we show that D-branes localized at arbitrary constant positions which were thought to be 1/4 supersymmetric in fact have sixteen supercharges whilst D-branes which were thought to be non-supersymmetric have eight supercharges. We discuss in detail the quantization in all cases.