Stochastic processes with independent increments on a groupare generalized to the concept of "white noise" on a Hopfalgebra or bialgebra. The main purpose of the book is thecharacterization of these processes as solutions of quantumstochastic differential equations in the sense of R.L.Hudsonand K.R. Parthasarathy. The notes are a contributionto quantum probability but they are also related toclassical probability, quantum groups, and operatoralgebras. The Az ma martingales appear as examples of whitenoise on a Hopf algebra which is a deformation of theHeisenberg group.The book will be of interest to probabilists and quantumprobabilists. Specialists in algebraic structures who arecurious about the role of their concepts in probablilitytheory as well as quantum theory may find the bookinteresting. The reader should havesome knowledge offunctional analysis, operator algebras, and probabilitytheory.