In non-commutative probability there are different notions of independence (free, Boolean and monotone), each with a notion of cumulants (analogue of classic cumulants) that linearize the addition of independent random variables. Formulas relating moments and cumulants can be expressed as a sum indexed by set partitions. Our goal is to construct a Hopf algebra T on non-crossing partitions NC that allows us to systematically study the transitions between distinct brands of cumulants in non-commutative probability. The Hopf algebra T is such that its character group can be identified with a group of 'semi-multiplicative' functions on the incidence algebra of NC, used to encode the formulas. While a basic tool of the Hopf Algebra, such as the antipode of T, helps in inverting such formulas. We will explain how T relates to other (more famous) Hopf algebras and explain some extensions we have worked on. This is a joint work with Celestino, Ebrahimi-Fard, Nica and Witzman.