  We study an anisotropic variant of the $d$-dimensional imperfect Bose gas,
where the asymptotic behaviour of the dispersion $\epsilon_{\bf k}$ at
vanishing momentum $\bf{k}$ may differ from the standard quadratic form. The
analysis reveals the key role of the shift exponent $\psi$ governing the
asymptotic behaviour of the critical temperature $T_c(\mu)$ as a function of
the chemical potential $\mu$ at $T_c\to 0$. We argue that the universality
classes of Bose-Einstein condensation admitted by the model may be classified
according to the allowed values of $\psi$ so that spatial dimensionality has
only an indirect impact on the transition properties. We analyse the
correlation function of the model and discuss its asymptotics depending on the
direction. Both for the perfect and imperfect anisotropic Bose gases, the
correlation function $\chi({\bf x})$ at $T>T_c$ turns out to show either
exponential decay or exponentially damped oscillatory behaviour depending on
the orientation of ${\bf x}$ with respect to the dispersion anisotropies.
