  In this paper, a Banach space framework is introduced in order to deal with
finite-dimensional path-dependent stochastic differential equations. A version
of Kolmogorov backward equation is formulated and solved both in the space of
$L^p$ paths and in the space of continuous paths using the associated
stochastic differential equation, thus establishing a relation between
path-dependent SDEs and PDEs in analogy with the classical case. Finally, it is
shown how to establish a connection between such Kolmogorov equation and the
analogue finite-dimensional equation that can be formulated in terms of the
path-dependent derivatives recently introduced by Dupire, Cont and Fourni\'{e}.
