  A submanifold of a pseudo-Riemannian manifold is said to have parallel mean
curvature vector if the mean curvature vector field H is parallel as a section
of the normal bundle. Submanifolds with parallel mean curvature vector are
important since they are critical points of some natural functionals.
  In this paper, we survey some classical and recent results on submanifolds
with parallel mean curvature vector. Special attention is paid to the
classification of space-like and Lorentz surfaces with parallel mean curvature
vector in Riemannian and indefinite space forms.
