  In this paper we study smooth toric Fano varieties using primitive relations
and toric Mori theory. We show that for any irreducible invariant divisor D in
a toric Fano variety X, we have $0\leq\rho_X-\rho_D\leq 3$, for the difference
of the Picard numbers of X and D. Moreover, if $\rho_X-\rho_D>0$ (with some
additional hypotheses if $\rho_X-\rho_D=1$), we give an explicit birational
description of X. Using this result, we show that when dim X=5, we have
$\rho_X\leq 9$.
  In the second part of the paper, we study equivariant birational morphisms f
whose source is Fano. We give some general results, and in dimension 4 we show
that f is always a composite of smooth equivariant blow-ups. Finally, we study
under which hypotheses a non-projective toric variety can become Fano after a
smooth equivariant blow-up.
