  We address the classification of ancient solutions to fully nonlinear
curvature flows for hypersurfaces. Under natural conditions on the speed of
motion we classify ancient solutions which are convex, noncollapsing, uniformly
two-convex and noncompact. There are exactly two possibilities -- every such
solution is either a self-similarly shrinking cylinder, or else is a
rotationally symmetric translating soliton. For a large class of flows this
yields a complete classification of the blow-up limits that can arise at a
singularity of a solution which is compact, embedded and two-convex.
