  Various cosmological models in frames of $F(T)$ gravity are considered. The
general scheme of constructing effective dark energy models with various
evolution is presented. It is showed that these models in principle are
compatible with $\Lambda$CDM model. The dynamics of universe governed by $F(T)$
gravity can mimics $\Lambda$CDM evolution in past but declines from it in a
future. We also construct some dark energy models with the "real"
(non-effective) equation-of-state parameter $w$ such that $w\leq-1$. It is
showed that in $F(T)$ gravity the Universe filled phantom field not necessarily
ends its existence in singularity. There are two possible mechanisms permitting
the final singularity. Firstly due to the nonlinear dependence between energy
density and $H^{2}$ ($H$ is the Hubble parameter) the universe can expands not
so fast as in the general relativity and in fact Little Rip regime take place
instead Big Rip. We also considered the models with possible bounce in future.
In these models the universe expansion can mimics the dynamics with future
singularity but due to bounce in future universe begin contracts.
