  We study a class of elliptic operators $A$ with unbounded coefficients
defined in $I\times\CR^d$ for some unbounded interval $I\subset\CR$. We prove
that, for any $s\in I$, the Cauchy problem $u(s,\cdot)=f\in C_b(\CR^d)$ for the
parabolic equation $D_tu=Au$ admits a unique bounded classical solution $u$.
This allows to associate an evolution family $\{G(t,s)\}$ with $A$, in a
natural way. We study the main properties of this evolution family and prove
gradient estimates for the function $G(t,s)f$. Under suitable assumptions, we
show that there exists an evolution system of measures for $\{G(t,s)\}$ and we
study the first properties of the extension of $G(t,s)$ to the $L^p$-spaces
with respect to such measures.
