This talk reviews how to describe ultracold trapped atomic gases within
both the canonical and the grand-canonical ensemble of statistical physics.
In the first part we develop a perturbative path integral approach for
calculating a recursion relation for the partition function of a fixed
number $N$ of weakly interacting bosons in different trap configurations.
With this we discuss how a two-particle $\delta$-interaction influences
the behaviour of the thermodynamic quantities near the quasi-critical point.
Furthermore, we show that the heat capacity and the number of particles
in the ground state, which defines the quasi-condensate, approach their
thermodynamic limits uniformly for all temperatures.
In the second part we analyse the ongoing Stuttgart experiment on the
Bose-Einstein condensation of chromium. Due to the diluteness of the gas,
we treat both the short-range, isotropic delta-interaction and the long-range,
anisotropic magnetic dipole-dipole interaction perturbatively with the help of
Feynman's diagrammatic technique of many-body theory.
We determine the shift of the critical temperature with respect to the purely
delta-interacting gas as a function of the relative orientation of the
symmetry axes of the trap and the atomic magnetic moments.
The difference of the critical temperatures between parallel and orthogonal
orientation of the symmetry axes only depends on the magnetic dipole-dipole
interaction and can be enhanced by increasing the number of chromium
atoms as well as the anisotropy of the trap.