Picard modular surfaces $X$, which arise classically as (compactifications of) the quotients of the unit ball in $\mathbb{C}^2$ 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 by describing the Albanese variety and its residual quotients, and then Lang's conjecture over finitely generated fields. It will then move on to discuss an ongoing, partially completed, program with M. Dimitrov to establish the paucity of rational points on the open part.