  A marginally trapped surface in a spacetime is a Riemannian surface whose
mean curvature vector is lightlike at every point. In this paper we give an
up-to-date overview of the differential geometric study of these surfaces in
Minkowski, de Sitter, anti-de Sitter and Robertson-Walker spacetimes. We give
the general local descriptions proven by Anciaux and his coworkers as well as
the known classifications of marginally trapped surfaces satisfying one of the
following additional geometric conditions: having positive relative nullity,
having parallel mean curvature vector field, having finite type Gauss map,
being invariant under a one-parameter group of ambient isometries, being
isotropic, being pseudo-umbilical. Finally, we provide examples of constant
Gaussian curvature marginally trapped surfaces and state some open questions.
