Around 2011, several research groups completed a project that was initiated by Lucien Hardy in 2001, but that dates back much further, both conceptually and technically: to reconstruct the Hilbert space formalism of quantum theory from simple, "operational" or "information-theoretic" principles. A role model for some of this research is Einstein's derivation of the Lorentz transformations from the relativity and light principles, giving a principled and structural explanation of an ad hoc collection of equations. This lecture will not discuss the conceptual or philosophical context of this research, but it intends to give a pedagogical summary of what has been achieved on the level of mathematical physics. The first idea to understand is that both quantum theory and classical probability theory can be seen as two special cases of generalized probabilistic theories (GPTs). There is a vast landscape of GPTs, and some of them would predict physical phenomena that are in some sense "even weirder than quantum". It now turns out that a few simple principles are enough to single out quantum theory from the landscape of GPTs (but with considerable mathematical effort!). I will summarize an example reconstruction, and sketch the larger context of this research within quantum foundations.