  In this paper we study the classification of ancient convex solutions to the
mean curvature flow in $\R^{n+1}$. An open problem related to the
classification of type II singularities is whether a convex translating
solution is $k$-rotationally symmetric for some integer $2\le k\le n$, namely
whether its level set is a sphere or cylinder $S^{k-1}\times \R^{n-k}$. In this
paper we give an affirmative answer for entire solutions in dimension 2. In
high dimensions we prove that there exist non-rotationally symmetric, entire
convex translating solutions, but the blow-down in space of any entire convex
translating solution is $k$-rotationally symmetric. We also prove that the
blow-down in space-time of an ancient convex solution which sweeps the whole
space $\R^{n+1}$ is a shrinking sphere or cylinder.
