We will discuss two different approaches to computing the L-functions of Shimura varieties associated with GU(2,1). Both approaches employ the comparison of the Grothendieck-Lefschetz formula with the Arthur-Selberg trace formula. The first approach, carried out by the author, takes as its starting point the recent work of Laumon and Morel. The second approach is due to Flicker. In both approaches, the principal challenge is that the Shimura varieties in question are non-compact, and one must use cohomology with compact supports. Time permitting, we will discuss the prospects for extending these approaches to the non-compact Shimura varieties associated with higher-rank unitary groups.