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 $${\mathsf {NP}}$$NP -complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is $${\mathsf {NP}}$$NP -complete on graphs with minimum degree two. In this paper, we show that, for any given constant $$c>1$$c > 1 , Matching Cut is $${\mathsf {NP}}$$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 $$\epsilon >0$$ϵ > 0 , Matching Cut remains $${\mathsf {NP}}$$NP -complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree $$\delta >n^{1-\epsilon }$$δ > n 1 - ϵ . We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree $$\delta \ge 3$$δ ≥ 3 in $$O^*(\lambda ^n)$$O ∗ ( λ n ) time, where $$\lambda$$λ is the positive root of the polynomial $$x^{\delta +1}-x^{\delta }-1$$x δ + 1 - x δ - 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 $$O^*(1.0099^n)$$O ∗ ( 1 . 0099 n ) on graphs with minimum degree $$\delta \ge 469$$δ ≥ 469 . Complementing our hardness results, we show that, for any two fixed constants $$1< c <4$$1 < c < 4 and $$c^{\prime }\ge 0$$c ′ ≥ 0 , Matching Cut is solvable in polynomial time for graphs with large minimum degree $$\delta \ge \frac{1}{c}n-c^{\prime }$$δ ≥ 1 c n - c ′ .