In this article we provide hardness results and approximation algorithms for the following three natural degree-constrained subgraph problems, which take as input an undirected graph G=(V,E). Let d≥2 be a fixed integer. The Maximumd−degree-bounded Connected Subgraph (MDBCS_d) problem takes as additional input a weight function ω:E→R⁺, and asks for a subset E^′⊆E such that the subgraph induced by E^′ is connected, has maximum degree at most d, and [Formula: see text] is maximized. The Minimum Subgraph of Minimum Degree≥d (MSMD_d) problem involves finding a smallest subgraph of G with minimum degree at least d. Finally, the Dual Degree-densek-Subgraph (DDDkS) problem consists in finding a subgraph H of G such that |V(H)|≤k and the minimum degree in H is maximized.