Metal nanoparticle arrays that support surface lattice resonances have emerged as an exciting platform for manipulating light-matter interactions at the nanoscale and enabling a diverse range of applications. Their recent prominence can be attributed to a combination of desirable photonic and plasmonic attributes: high electromagnetic field enhancements extended over large volumes with long-lived lifetimes. This Review will describe the design rules for achieving high-quality optical responses from metal nanoparticle arrays, nanofabrication advances that have enabled their production, and the theory that inspired their experimental realization. Rich fundamental insights will focus on weak and strong coupling with molecular excitons, as well as semiconductor excitons and the lattice resonances. Applications related to nanoscale lasing, solid-state lighting, and optical devices will be discussed. Finally, prospects and future open questions will be described.