  First, we revisit functional It\^o/path-dependent calculus started by B.
Dupire, R. Cont and D.-A. Fourni\'e, using the formulation of calculus via
regularization. Relations with the corresponding Banach space valued calculus
introduced by C. Di Girolami and the second named author are explored. The
second part of the paper is devoted to the study of the Kolmogorov type
equation associated with the so called window Brownian motion, called
path-dependent heat equation, for which well-posedness at the level of
classical solutions is established. Then, a notion of strong approximating
solution, called strong-viscosity solution, is introduced which is supposed to
be a substitution tool to the viscosity solution. For that kind of solution, we
also prove existence and uniqueness. The notion of strong-viscosity solution
motivates the last part of the paper which is devoted to explore this new
concept of solution for general semilinear PDEs in the finite dimensional case.
We prove an equivalence result between the classical viscosity solution and the
new one. The definition of strong-viscosity solution for semilinear PDEs is
inspired by the notion of "good" solution, and it is based again on an
approximating procedure.
