Picard modular surfaces X, which arise classically as (compactifications of) the quotients of the unit ball in C^2 by arithmetic lattices Gamma in SU(2, 1), have mirific properties, making them a crucial place to test various arithmetic and geometric conjectures. This talk will begin describing the albanese variety of X, which the speaker had earlier worked on with Kumar Murty, and then move on to discuss its utility in the study of periods and rational points in an ongoing, partially completed, program with M. Dimitrov.