  We describe basic motivations behind quantum or noncommutative probability,
introduce quantum L\'evy processes on compact quantum groups, and discuss
several aspects of the study of the latter in the example of quantum
permutation groups. The first half of this paper is a survey on quantum
probability, compact quantum groups, and L\'evy processes on compact quantum
groups. In the second half the theory is applied to quantum permutations
groups. Explicit examples are constructed and certain classes of such L\'evy
processes are classified.
