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Einstein's classic description of non-interacting bosons predicts at what temperature the phase transition to Bose-Einstein condensation occurs, but how this critical temperature changes when the bosons start to interact weakly is an old problem. To study this question, we analyse a variational reformulation of the Bogoliubov approximation that is commonly used to describe weakly-interacting Bose gases. I will describe several aspects of the resulting model: its derivation, the existence of minimizers, the phase diagram, and finally the critical temperature, which has the form predicted by Lee and Yang in 1958 and a constant that agrees with numerical simulations.