Lead-halide perovskites have demonstrated astonishing increases in power conversion efficiency in photovoltaics over the last decade. The most efficient perovskite devices now outperform industry-standard multi-crystalline silicon solar cells, despite the fact that perovskites are typically grown at low temperature using simple solution-based methods. However, the toxicity of lead and its ready solubility in water are concerns for widespread implementation. These challenges, alongside the many successes of the perovskites, have motivated significant efforts across multiple disciplines to find lead-free and stable alternatives which could mimic the ability of the perovskites to achieve high performance with low temperature, facile fabrication methods. This Review discusses the computational and experimental approaches that have been taken to discover lead-free perovskite-inspired materials, and the recent successes and challenges in synthesizing these compounds. The atomistic origins of the extraordinary performance exhibited by lead-halide perovskites in photovoltaic devices is discussed, alongside the key challenges in engineering such high-performance in alternative, next-generation materials. Beyond photovoltaics, this Review discusses the impact perovskite-inspired materials have had in spurring efforts to apply new materials in other optoelectronic applications, namely light-emitting diodes, photocatalysts, radiation detectors, thin film transistors and memristors. Finally, the prospects and key challenges faced by the field in advancing the development of perovskite-inspired materials towards realization in commercial devices is discussed.