The interplay between probability theory and number theory has a rich history of producing deep results and conjectures. Important instances are the works of Erdös, Kac, Selberg, Montgomery, Soundararajan and Granville, to name a few. This talk will review recent results in this spirit where the insights of probability, of branching processes in particular, have led to a better understanding of large values of the Riemann zeta function on the critical line.