The computational difficulty of solving quantum many-body problems is a significant obstacle to understanding central questions in quantum condensed-matter physics and in quantum chemistry. This talk will introduce how arrays of trapped atomic ions can be engineered and reprogrammed to emulate the behavior of interesting materials and quantum-chemical systems. First, I will review some of the groundbreaking experiments to explore the physics of interacting spin systems using 1D chains of ions. Then, I will describe our creation of a 2D trapped ion array, which significantly expands the classes of quantum matter that can be emulated. We characterize the structural phase boundaries and vibrational mode frequencies of ions in this geometry, as well as the heating rates parallel and perpendicular to the 2D crystal plane. We find that these 2D crystals will serve as a robust platform for future experiments investigating strongly-correlated matter and quantum nuclear chemical dynamics.