  In this survey, we will focus on the mean curvature flow theory with sphere
theorems, and discuss the recent developments on the convergence theorems for
the mean curvature flow of arbitrary codimension inspired by the Yau rigidity
theory of submanifolds. Several new differentiable sphere theorems for
submanifolds are obtained as consequences of the convergence theorems for the
mean curvature flow. It should be emphasized that Theorem 4.1 is an optimal
convergence theorem for the mean curvature flow of arbitrary codimension, which
implies the first optimal differentiable sphere theorem for submanifolds with
positive Ricci curvature. Finally, we present a list of unsolved problems in
this area.
