The global ocean is a significant sink for anthropogenic carbon (C_ant), absorbing roughly a third of human CO₂ emitted over the industrial period. Robust estimates of the magnitude and variability of the storage and distribution of C_ant in the ocean are therefore important for understanding the human impact on climate. In this synthesis we review observational and model-based estimates of the storage and transport of C_ant in the ocean. We pay particular attention to the uncertainties and potential biases inherent in different inference schemes. On a global scale, three data-based estimates of the distribution and inventory of C_ant are now available. While the inventories are found to agree within their uncertainty, there are considerable differences in the spatial distribution. We also present a review of the progress made in the application of inverse and data assimilation techniques which combine ocean interior estimates of C_ant with numerical ocean circulation models. Such methods are especially useful for estimating the air–sea flux and interior transport of C_ant, quantities that are otherwise difficult to observe directly. However, the results are found to be highly dependent on modeled circulation, with the spread due to different ocean models at least as large as that from the different observational methods used to estimate C_ant. Our review also highlights the importance of repeat measurements of hydrographic and biogeochemical parameters to estimate the storage of C_ant on decadal timescales in the presence of the variability in circulation that is neglected by other approaches. Data-based C_ant estimates provide important constraints on forward ocean models, which exhibit both broad similarities and regional errors relative to the observational fields. A compilation of inventories of C_ant gives us a "best" estimate of the global ocean inventory of anthropogenic carbon in 2010 of 155 ± 31 PgC (±20% uncertainty). This estimate includes a broad range of values, suggesting that a combination of approaches is necessary in order to achieve a robust quantification of the ocean sink of anthropogenic CO₂.