  This article is on the research of Wilhelm von Waldenfels in the mathematical
field of quantum (or non-commutative) probability theory. Wilhelm von
Waldenfels was one of the pioneers, even one of the founders, of quantum
probability. We concentrate on a small part of his scientific work. The aspects
of physics are practically not mentioned at all. There is nothing on his
results in classical probability on groups (Waldenfels operators). This is an
attempt to show how the concepts of non-commutative notions of independence and
of L\'evy processes on structures like Hopf algebras developed from the ideas
of Wilhelm von Waldenfels.
