  Neutral atoms become strongly interacting when their electrons are excited to
loosely bound Rydberg states. We investigate the strongly correlated quantum
phases of matter that emerge in two-dimensional atom arrays where three Rydberg
levels are used to encode an effective spin-1 degree of freedom. Dipolar
exchange between such spin-1 Rydberg atoms naturally yields two distinct
models: (i) a two-species hardcore boson model, and (ii) upon tuning near a
F\"orster resonance, a dipolar spin-1 XY model. Through extensive, large-scale
infinite density matrix renormalization group calculations, we provide a broad
roadmap predicting the quantum phases that emerge from these models on a
variety of lattice geometries: square, triangular, kagome, and ruby. We
identify a wealth of correlated states, including lattice supersolids and
simplex phases, all of which can be naturally realized in near-term
experiments.
