In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Finding cuts of maximum size is one of one of Karp's 21 NP-complete problems, and approximation algorithms for this problem have inspired many fundamental directions in approximation algorithm design. However, very little is known about the parameterized complexity of the Max-Cut problem. We present new combinatorial and fixed-parameter tractability results for Max-Cut and related problems. Some of our results answer long-standing open questions from parameterized complexity or asymptotically improve earlier results.