Over the past few years, there has been growing evidence, at the heuristic, the mathematically rigorous, and even the experimental level, that models of one-dimensional interface growth exhibit a "universal" behaviour at large scales. More precisely, it is conjectured that there exists a self-similar space-time process called the "KPZ fixed point" which attracts a very large class of microscopic models under suitable rescaling. It has also emerged that a certain ill-posed nonlinear stochastic PDE, the KPZ equation, has a "weak universality" property in the sense that large classes of models with a tuneable parameter converge to its solutions at intermediate scalings in the limit where the tuneable parameter is small. I will review some of the existing mathematical results supporting these conjectures and give an idea of the mathematical techniques involved in their proofs.