  We review the theory of vortices in trapped dilute Bose-Einstein condensates
and compare theoretical predictions with existing experiments. Mean-field
theory based on the time-dependent Gross-Pitaevskii equation describes the main
features of the vortex states, and its predictions agree well with available
experimental results. We discuss various properties of a single vortex,
including its structure, energy, dynamics, normal modes and stability, as well
as vortex arrays. When the nonuniform condensate contains a vortex, the
excitation spectrum includes unstable (``anomalous'') mode(s) with negative
frequency. Trap rotation shifts the normal-mode frequencies and can stabilize
the vortex. We consider the effect of thermal quasiparticles on vortex normal
modes as well as possible mechanisms for vortex dissipation. Vortex states in
mixtures and spinor condensates are also discussed.
