Of fundamental importance in number theory is the question of non-vanishing of central L-values of L-functions. One approach, explained in the talk, is to make use of the Relative trace formula (which will be introduced from scratch); a basic example of interest involves twists of L-functions of classical modular forms. If time permits, we will explain the recent work with Michel and Yang on $U(2)$-twists of $U(3)$ L-functions.