Abstract: : We will discuss the decay of a metastable vacuum catalysed by black holes. The phenomenological interest in this problem lies in the fact that our current electroweak vacuum may not be absolutely stable. Its lifetime, while sufficiently large in the present-day Universe, may be significantly reduced in extreme environments such as the vicinity of small hot primordial black holes, which could be produced abundantly in the early Universe. From the theoretical point of view, the challenge is to compute the rate of decay of the false quantum vacuum state which lives in a curved spacetime and which is generally out of thermal equilibrium. I will discuss a general method of doing so and illustrate it using toy models of dilaton black holes in two dimensions. We will see that different vacuum states associated with a black hole have parametrically different lifetimes. Hence, in computing the enhancement of the electroweak vacuum decay rate near black holes it is important to choose the physically relevant (Unruh) vacuum state. Finally, I will outline what remains to be done in the realistic case of black holes in four dimensions.