Neutrino Masses

This is a review for Reports of Progress in Physics. After an introduction we start by explaining the different neutrino masses corresponding to different types of neutrinos, Dirac or Majorana, in section 2. In section 3 we discuss the main elementary particle models for neutrino masses and their distinctive phenomenological consequences. In section 4 we describe the status of direct mass searches and Majorana mass searches in neutrinoless double beta decays. In section 5 we go over the many cosmological implications of, and constraints on, neutrino properties, mainly masses and lifetimes. Sections 6, 7 and 8 review neutrino oscillations, the solar neutrino problem and the atmospheric neutrino problem, their implications and the current and future experiments. In particular, we explain oscillations in vacuum in section 6 and oscillations in matter in section 7. Section 9 summarizes the main bounds imposed by stars, mainly SN1987A. A few concluding remarks follow.