In this talk, we discuss recent developments in ancient solutions to the mean curvature flow in
higher dimensions. Consider an ancient flow asymptotic to a cylinder with the number of R factors equal
to k, we show that the asymptotic behavior of the flow is characterized by a k x k matrix Q whose
eigenvalues can only be 0 and 1. We further discuss the cases where Q is fully degenerate or fully
nondegenerate under the noncollapsing assumption. In the fully degenerate case, we obtain a complete
classification. In the fully nondegenerate case, we establish a rigidity result showing that the solutions are
determined by only k-1 parameters. This is based on joint work with Beomjun Choi and Wenkui Du.