1 Introduction.- 2 Preliminaries on Lie groups.- 3 Hopf algebras, quantum groups and braided spaces.- 4 Stochastic processes on quantum groups.- 5 Markov structure of quantum Lévy processes.- 6 Diffusions on braided spaces.- 7 Evolution equations and Lévy processes on quantum groups.- 8 Gauss laws in the sense of Bernstein on quantum groups.- 9 Phase retrieval for probability distributions on quantum groups and braided groups.- 10 Limit theorems on quantum groups.