  In this note, we discuss the mean curvature flow of graphs of maps between
Riemannian manifolds. Special emphasis will be placed on estimates of the flow
as a non-linear parabolic system of differential equations. Several global
existence theorems and applications to isotopy problems in geometry and
topology will be presented. The results are based on joint works of the author
with his collaborators I. Medo\v{s}, K. Smoczyk, and M.-P. Tsui.
