The Plateau problem consists in finding the least area surface spanning a given contour. This classical question, already posed by Lagrange around 1760, has motivated many developments in analysis, mathematical physics and geometry in the last centuries. In this colloquium I will revisit some of the main issues and achievements in the solution of this problem, focusing mostly on its analytical aspects. In particular, I will explain why in general the solutions to the Plateau problem cannot be a regular surface and I will survey some known results and open questions about the singularities of minimal surfaces up to very recent developments.