Free probability theory, a species of non-commutative probability theory, is amazing for several reasons. Not only has it nice combinatorial features underlying it but also profound connections with other fields, in particular physics. Recently, we established a priori unexpected relations with some very prominent algebraic objects, in particular Hopf algebras. In this talk we will carefully introduce some of the basic features and give a glance at future directions.

In this talk we derive a novel and fundamental relation between C*-algebras and Complex Cobordism. We endow the space of complex genera with the structure of an infinite Lie group, with the underlying manifold being isomorphic to the group of automorphisms of the formal disc, and relate it then to operator algebras. The group operation introduced can be understood combinatorially in terms of planar rooted trees. Finally we shall give connections with Conformal Field Theory.