  We investigate cosmological scenarios in the theory of gravity with the
scalar field possessing a non-minimal kinetic coupling to the curvature. It is
shown that the kinetic coupling provides an essentially new inflationary
mechanism. Namely, at early cosmological times the domination of coupling terms
in the field equations guarantees the quasi-De Sitter behavior of the scale
factor: $a(t)\propto e^{H_{\kappa} t}$ with $H_\kappa=1/\sqrt{9\kappa}$, where
$\kappa\simeq 10^{-74}$ sec$^2$ is the coupling parameter. The primary
inflationary epoch driven by non-minimal kinetic coupling comes to the end at
$t_f \simeq 10^{-35}$ sec. Later on, the matter terms are dominating, and the
universe enters into the matter-dominated epoch which lasts approximately
$0.5H_0^{-1}\sim 0.5\times10^{18}$ sec. Then, the cosmological term comes into
play, and the universe enters into the secondary inflationary epoch with
$a(t)\propto e^{H_{\Lambda} t}$, where $H_\Lambda=\sqrt{\Lambda/3}$. Note that
the present value of the acceleration parameter $q=\ddot a a/\dot a^2$ is
estimated as $q_0\simeq0.25$, that is the universe is at the beginning of the
epoch of accelerated expansion. Thus, the cosmological model non-minimal
kinetic coupling represents the realistic cosmological scenario which
successfully describes basic cosmological epochs and provide the natural
mechanism of epoch change without any fine-tuned potential.
