The main theme of this expository paper is the relation between analysis and probability in the context of diffusion theory. Section 1 discusses in rather heuristic fashion the very satisfying solution to the problem of describing diffusion processes which Kolmogorov achieved via PDE theory (the theory of partial differential equations) and his criterion for path continuity. Section 2 describes how Ito calculus totally transformed the subject by allowing us to construct the sample paths of a diffusion process X by solving an SDE (stochastic differential equation) driven by brownian motion. (Of course, SDES have great intrinsic importance too as noisy perturbations of nonlinear dynamical systems.) Though section2 begins heuristically, the mathematics is then tightened up. This paper is, after all, a tribute to the man whose greatest contribution to science is his setting probability theory on a rigorous foundation. Once Kolmogorov's precise language is available, section2 then takes a quick sight-seeing trip through some of the great developments by Doob, Ito and their successors. (In an age in which so many do simulations of Ito equations, I have explained precisely in the briefest possible fashion what the exact theory is. It is easy and usable.) You will just have time for a snapshot of how brownian motion on the orthonormal frame bundle is linked to index theorems, and of what the Malliavin calculus is about. You will, however, be advised on guide-books on these and other areas (including physicists' favourites: large deviations, measure-valued diffusions, etc.), so that you can later explore at your leisure. Confession: the paper consists very largely of selected tracks (remixed!) from the album (Rogers & Williams 1987 Diffusions, Markov processes and martingales; Chichester: Wiley). Tributes to Kolmogorov's work in probability and statistics have appeared in (every book ever written on probability and in) Ann. Probability 17 (1989), 815-964, Ann. Statist. 18 (1990), 987-1031, Bull. Lond. math. Soc. 22 (1990), 31-100, Teor. Veroyatnost Primenen 34 (1989), no. 1, Usp. mat. Nauk 43 (1988), no. 6. The official biography by Shiryaev will be a wonderful volume. For me, this paper is a further expression of my thanks to Kolmogorov and (as he would have wished) to Levy, Doob and Ito too.