Random walks on groups and free convolutions

We start with a result of Harry Kesten stating that a symmetric random walk on the free group, F_d, of order d is transient for d>1. I will sketch a proof that leads us to the concept of free independence of non-commutative random variables as introduced by Dan Voiculescu. A main tool in the proof is the free additive convolution of probability measures, an operation associated with the addition of freely independent random variables. Kesten's result follows from a special case of the free convolution which is accessible via explicit calculations. I will then discuss properties of the free convolution of generic probability measures and present some recent regularity results. I will conclude by establishing a link with random matrix theory and explain the role of free probability in some on-going research projects.