  The study of the mean curvature flow from the perspective of partial
differential equations began with Gerhard Huisken's pioneering work in 1984.
Since that time, the mean curvature flow of hypersurfaces has been a lively
area of study. Although Huisken's seminal paper is now just over twenty-five
years old, the study of the mean curvature flow of submanifolds of higher
codimension has only recently started to receive attention. The mean curvature
flow of submanifolds is the main object of investigation in this thesis, and
indeed, the central results we obtain can be considered as high codimension
analogues of some early hypersurface theorems. The result of Huisken's 1984
paper roughly says that convex hypersurfaces evolve under the mean curvature
flow to round points in finite time. Here we obtain the result that if the
ratio of the length of the second fundamental form to the length of the mean
curvature vector is bounded (by some explicit constant depending on dimension
but not codimension), then the submanifold will evolve under the mean curvature
flow to a round point in finite time. We investigate evolutions in flat and
curved backgrounds, and explore the singular behaviour of the flows as the
first singular time is approached.
