  The mean curvature flow is an evolution process under which a submanifold
deforms in the direction of its mean curvature vector. The hypersurface case
has been much studied since the eighties. Recently, several theorems on
regularity, global existence and convergence of the flow in various ambient
spaces and codimensions were proved. We shall explain the results obtained as
well as the techniques involved. The potential applications in symplectic
topology and mirror symmetry will also be discussed.
