Arrays of coupled superconducting qubits are a compelling platform for analog quantum simulations of solid-state matter and many-body physics. These devices natively emulate the Bose-Hubbard model while offering a high degree of control, fast operation rates, and site-resolved readout. We discuss recent experiments using a 4-by-4 array of transmon qubits. By adopting a parametric coupling scheme, we emulate a 2D material in an adjustable electromagnetic field and demonstrate the Aharonov-Bohm effect, Faraday’s law of induction, and the Hall effect. We then emulate a material with flat bands, and we study the interplay between band structure and disorder-induced localization and delocalization. We conclude by discussing a near-term path towards utility-scale analog quantum simulators based on superconducting qubit arrays.