One of the key aspects of black hole physics is what have become known as the black hole no-hair theorems: stationary, asymptotically flat, electrovac black holes are completely characterised by the Kerr-Newman solution. In this talk I will explain how the no-hair theorems extend when a scalar field is coupled to gravity, and present a recent result ruling out hairy black holes with a non-canonical scalar field. This result applies to a wide class of scalar field theories often considered in cosmology, and also rules out new rotating black holes with this matter source. Time permitting, I will also discuss whether stationary black holes can admit time-dependent scalar hair.