Introduction: Eric Charpentier, Annick Lesne, Nikolaï Nikolski .- The youth of Andrei Nikolaevich and Fourier series: Jean-Pierre Kahane .- Kolmogorov's contribution to intuitionistic logic: Thierry Coquand.- Some aspects of the probabilistic work: Loïc Chaumont, Laurent Mazliak, Marc Yor.- Infinite dimensional Kolmogorov equations: Giuseppe Da Prato.- From Kolmogorov's theorem on empirical distribution to number theory: Kevin Ford.- Kolmogorov's -entropy and the problem of statistical estimation: Mikhail Nikouline, Valentin Solev.- Kolmogorov and topology: Victor M. Buchstaber .- Geometry and approximation theory in A. N. Kolmogorov's works: Vladimir M. Tikhomirov.- Kolmogorov and population dynamics: Karl Sigmund.- Resonances and small divisors: Etienne Ghys.- The KAM Theorem: John H. Hubbard .-From Kolmogorov's Work on Entropy of Dynamical Systems to Non-uniformly Hyperbolic Dynamics: Denis V. Kosygin, Yakov G. Sinai.- From Hilbert's 13th Problem to the theory of neural networks: constructive aspects of Kolmogorov's Superposition Theorem: Vasco Brattka .- Kolmogorov Complexity: Bruno Durand, Alexander Zvonkin.- Algorithmic Chaos and the Incompressibility Method: Paul Vitanyi.