An instance of the Connected Maximum Cut problem consists of an undirected graph G=(V,E) and the goal is to find a subset of vertices S⊆V that maximizes the number of edges in the cut δ(S) such that the induced graph G[S] is connected. We present the first non-trivial Ω([Formula presented]) approximation algorithm for the Connected Maximum Cut problem in general graphs using novel techniques. We then extend our algorithm to edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in contrast to the classical Max-Cut problem that can be solved in polynomial time on planar graphs, we show that the Connected Maximum Cut problem remains NP-hard on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the Connected Maximum Cut problem on planar graphs and more generally on bounded genus graphs.