  A well-known question of Perelman concerns the classification of noncompact
ancient solutions to the Ricci flow in dimension $3$ which have positive
sectional curvature and are $\kappa$-noncollapsed. In this paper, we solve the
analogous problem for mean curvature flow in $\mathbb{R}^3$, and prove that the
rotationally symmetric bowl soliton is the only noncompact ancient solution of
mean curvature flow in $\mathbb{R}^3$ which is strictly convex and
noncollapsed.
