One of the fundamental problems in ergodic theory is to establish pointwise convergence almost everywhere for ergodic averages. During this talk we will discuss recent developments in pointwise ergodic theory associated with polynomial averages in the nilpotent case. We will report on recent progress towards establishing the famous Furstenberg-Bergelson-Leibman conjecture in the noncommutative situation. Based on joint work with A.D. Ionescu, Á. Magyar and M. Mirek.