We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group G_{K}$G_{K}$, K$K$ a finite extension of \mathbb{Q}_{p}$\mathbb{Q}_{p}$, for any p>2$p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that K/\mathbb{Q}_{p}$K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture.