In this thesis, we analyse a variational reformulation of the Bogoliubov approximation that is used to
describe weakly-interacting translationally-invariant Bose gases. For the resulting model, the `Bogoliubov
free energy functional', we demonstrate existence of minimizers as well as the presence of a phase
transition to Bose{Einstein condensation, and establish the phase diagram. We also give a calculation of
the critical temperature assuming the gas is dilute, and nd that it agrees with earlier numerical studies.
The thesis contains an introduction, a physical review paper outlining the main results and ideas,
and two mathematical papers with detailed proofs