  Let $(X,f_{1,\infty})$ be a nonautonomous dynamical system. In this paper we
summarize known definitions of periodic points for general nonautonomous
dynamical systems and propose a new, very natural, definition of asymptotic
periodicity. Moreover, this definition is resistant to changes of a beginning
of the sequence generating the nonautonomous system. We show the relations
among these definitions and discuss their properties. We prove that for
pointwise convergent nonautonomous systems topological transitivity together
with dense set of asymptotically periodic points imply sensitivity. We also
show that event for unoformly convergent systems the nonautonomous analog of
Sharkovsky's Theorem is not valid for most definitions of periodic points.
