In recent years, observations of the Sunyaev-Zeldovich (SZ) effect have had significant cosmological implications and have begun to serve as a powerful and independent probe of the warm and hot gas that pervades the Universe. As a few pioneering studies have already shown, SZ observations both complement X-ray observations -- the traditional tool for studying the intra-cluster medium -- and bring unique capabilities for probing astrophysical processes at high redshifts and out to the low-density regions in the outskirts of galaxy clusters. Advances in SZ observations have largely been driven by developments in centimetre-, millimetre-, and submillimetre-wave instrumentation on ground-based facilities, with notable exceptions including results from the Planck satellite. Here we review the utility of the thermal, kinematic, relativistic, non-thermal, and polarised SZ effects for studies of galaxy clusters and other large scale structures, incorporating the many advances over the past two decades that have impacted SZ theory, simulations, and observations. We also discuss observational results, techniques, and challenges, and aim to give an overview and perspective on emerging opportunities, with the goal of highlighting some of the exciting new directions in this field.
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