The total mass of a galaxy cluster is one of its most fundamental properties. Together with the redshift, the mass links observation and theory, allowing us to use the cluster population to test models of structure formation and to constrain cosmological parameters. Building on the rich heritage from X-ray surveys, new results from Sunyaev-Zeldovich and optical surveys have stimulated a resurgence of interest in cluster cosmology. These studies have generally found fewer clusters than predicted by the baseline Planck LCDM model, prompting a renewed effort on the part of the community to obtain a definitive measure of the true cluster mass scale. Here we review recent progress on this front. Our theoretical understanding continues to advance, with numerical simulations being the cornerstone of this effort. On the observational side, new, sophisticated techniques are being deployed in individual mass measurements and to account for selection biases in cluster surveys. We summarise the state of the art in cluster mass estimation methods and the systematic uncertainties and biases inherent in each approach, which are now well identified and understood, and explore how current uncertainties propagate into the cosmological parameter analysis. We discuss the prospects for improvements to the measurement of the mass scale using upcoming multi-wavelength data, and the future use of the cluster population as a cosmological probe.
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