Swarming is a collective flagella-dependent movement of bacteria across a surface that is observed across many species of bacteria. Due to the prevalence and diversity of this motility modality, multiple models of swarming have been proposed, but a consensus on a general mechanism for swarming is still lacking. Here, we focus on swarming by Pseudomonas aeruginosa due to the abundance of experimental data and multiple models for this species, including interpretations that are rooted in biology and biophysics. In this review, we address three outstanding questions about P. aeruginosa swarming: what drives the outward expansion of a swarm, what causes the formation of dendritic patterns (tendrils), and what are the roles of flagella? We review models that propose biologically active mechanisms including surfactant sensing as well as fluid mechanics-based models that consider swarms as thin liquid films. Finally, we reconcile recent observations of P. aeruginosa swarms with early definitions of swarming. This analysis suggests that mechanisms associated with sliding motility have a critical role in P. aeruginosa swarm formation.