  We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe
arXiv:hep-th/9405096 considering several examples of matrix model
$\tau$-functions with natural genus expansion. Among the examples there are
solutions of KP equations of special interest, such as generating function for
simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and
Brezin-Gross-Witten model. We show that all these models with parameter $\hbar$
are $\tau$-functions of the $\hbar$-KP hierarchy and the expansion in $\hbar$
for the $\hbar$-KP coincides with the genus expansion for these models.
Furthermore, we show a connection of recent papers considering the
$\hbar$-formulation of the KP hierarchy arXiv:1509.04472, arXiv:1512.07172 with
original Takasaki-Takebe approach. We find that in this approach the recovery
of enumerative geometric meaning of $\tau$-functions is straightforward and
algorithmic.
