  In this paper we investigate the epochs in which the Universe started
accelerating and when it began to become dark energy-dominated (i.e., the
dynamics of the expansion of the Universe dominated by the dark energy). We
provide analytic expressions to calculate the redshifts of these epochs as a
function of density parameters. Moreover, we review and discuss cosmological
models with a dark energy component, which can have an interesting
characteristic, namely, they never stop accelerating. This holds even if the
Universe is at present time either flat, open, or closed. If the dark energy is
the cosmological constant the Universe will eventually end up undergoing an
exponentially expansion phase, and the total density parameter converging to
$\Omega=1$. This is exactly what is considered in inflationary scenario to
generate the initial conditions for the big bang. One can then argue that the
Universe begun with an inflationary phase and will end up with another
inflationary phase. Thus, it follows that in both the early and the late
Universe $\Omega \to 1$. We also discuss the above issues in the context of the
XCDM parametrization.
