Summary form only given, as follows. The nature and effects of plasma oscillations and instabilities in plasma thrusters will be reviewed in the context of the Hall thruster and the magnetoplasmadynamic thruster (MPDT). We will review the relevant experimental data and related analytical and numerical work. We will focus on the role of plasma instabilities and the associated turbulence in regulating and enhancing transport in the plasmas of these devices. In particular, the anomalous resistivity effect in MPDTs and the anomalous cross-field electron mobility in Hall thrusters will be illustrated.