In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be {mathsf {NP}}-complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is {mathsf {NP}}-complete on graphs with minimum degree two. In this paper, we show that, for any given constant c>1, Matching Cut is {mathsf {NP}}-complete in the class of graphs with minimum degree c and this restriction of Matching Cut has no subexponential-time algorithm in the number of vertices unless the Exponential-Time Hypothesis fails. We also show that, for any given constant epsilon >0, Matching Cut remains {mathsf {NP}}-complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree delta >n^{1-epsilon }. We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree delta ge 3 in O^*(lambda ^n) time, where lambda is the positive root of the polynomial x^{delta +1}-x^{delta }-1. Despite the hardness results, this is a very fast exact exponential-time algorithm for Matching Cut on graphs with large minimum degree; for instance, the running time is O^*(1.0099^n) on graphs with minimum degree delta ge 469. Complementing our hardness results, we show that, for any two fixed constants 1< c <4 and c^{prime }ge 0, Matching Cut is solvable in polynomial time for graphs with large minimum degree delta ge frac{1}{c}n-c^{prime }.