Scalar-tensor theories of gravity provide an intriguing and compelling explanation to the dark energy problem. They have received increased attention in recent years thanks to a wealth of developments both in the theoretical and experimental sides. The class of models known as “degenerate” provide a particularly interesting case. These theories extend general relativity by a single degree of freedom, despite their equations of motion being higher than second order, a virtue made possible by the existence of an additional constraint that removes the would-be instability associated to a ghost. This constraint can however be obstructed by matter fields, even when minimally coupled to the metric. In this talk I will present this issue in detail, explaining through some illustrative examples the precise ways in which the extra degree of freedom may reappear. This occurs in the Hamiltonian language through a loss of constraints, which may happen either when the kinetic matrix is not block-diagonal in the presence of matter fields, or when the matter sector itself has constraints. I will next turn to the more physically relevant case of fermionic matter, and show that spin-1/2 fermions evade these issues and can thus be consistently coupled to degenerate theories of scalar-tensor gravity.