We derive a functional change of variable formula for {\it non-anticipative}
functionals defined on the space of right continuous paths with left limits.
The functional is only required to possess certain directional derivatives,
which may be computed pathwise. Our results lead to functional extensions of
the Ito formula for a large class of stochastic processes, including
semimartingales and Dirichlet processes. In particular, we show the stability
of the class of semimartingales under certain functional transformations.
Keywords: change of variable formula, functional derivative, functional
calculus, stochastic integral, stochastic calculus, quadratic variation, Ito
formula, Dirichlet process, semimartingale, Wiener space, F\"ollmer integral,
Ito integral, cadlag functions.