Path-dependent Kolmogorov equations are a class of infinite dimensional partial differential equations on the space of cadlag functions which extend Kolmogorov’s backward equation to path-dependent functionals of stochastic processes. Solutions of such equations are non-anticipative functionals which extend the notion of harmonic function to a non-Markovian, functional setting. We discuss existence, uniqueness and properties of weak and strong solutions of path-dependent Kolmogorov equations using the Functional Ito calculus. Time permitting, some applications to mathematical finance and non-Markovian stochastic control will be discussed.