Abstract: Consider a compact manifold M of dimension at least 2 and the space of $C^r$-smooth diffeomorphisms Diff$^r(M)$. The
classical Artin-Mazur theorem says that for a dense subset D
of Diff$^r(M)$ the number of isolated periodic points grows at
most exponentially fast (call it the A-M property).
We extend this result and prove that diffeomorphisms having
only hyperbolic periodic points with the A-M property are dense
in Diff$^r(M)$. Our proof of this result is much simpler than
the original proof of Artin-Mazur.
The second main result is that the A-M property is not (Baire)
generic. Moreover, in a Newhouse domain ${\cal N} \subset
\textup{Diff}^r(M)$, an arbitrary quick growth of the number of
periodic points holds on a residual set. This result follows
from a theorem of Gonchenko-Shilnikov-Turaev, a detailed proof
of which is also presented.