  We review recent advances in the theory of the three-dimensional dilute
homogeneous Bose gas at zero and finite temperature. Effective field theory
methods are used to formulate a systematic perturbative framework that can be
used to calculate the properties of the system at T=0. The perturbative
expansion of these properties is essentially an expansion in the gas parameter
$\sqrt{na^3}$, where $a$ is the s-wave scattering length and $n$ is the number
density. In particular, the leading quantum corrections to the ground state
energy density, the condensate depletion, and long-wavelength collective
excitations are rederived in and efficient and economical manner. We also
discuss nonuniversal effects. These effects are higher-order corrections that
depend on properties of the interatomic potential other than the scattering
length, such as the effective range. We critically examine various approaches
to the dilute Bose gas in equilibrium at finite temperature. These include the
Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov
approximation, the $\Phi$-derivable approach, optimized perturbation theory,
and renormalization group techniques. Finally, we review recent calculations of
the critical temperature of the dilute Bose gas, which include 1/N-techniques,
lattice simulations, self-consistent calculations, and variational perturbation
theory.
