In support of projects for forecasting and mitigating the deleterious effects of extreme space-weather storms, an examination is made of the intensities of magnetic superstorms recorded in the Dst index time series (1957-2016). Modified peak-over-threshold and solar-cycle, block-maximum sampling of the Dst time series are performed to obtain compi-lations of storm-maximum −Dstm intensity values. Lognormal, upper-limit lognormal, generalized Pareto, and generalized extreme-value model distributions are fitted to the−Dstm data using a maximum-likelihood algorithm. All four candidate models provide good representations of the data. Comparisons of the statistical significance and good-ness of fits of the various models gives no clear indication as to which model is best. The statistical models are used to extrapolate to extreme-value intensities, such as would be expected (on average) to occur once per century. An upper-limit lognormal fit to peak-over-threshold −Dstm data above a superstorm threshold of 283 nT gives a 100-year ex-trapolated intensity of 542 nT and a 68% confidence interval (obtained by bootstrap re-sampling) of [466, 583] nT. An upper-limit lognormal fit to solar-cycle, block-maximum−DstBM data gives a 9-solar-cycle (approximately 100-year) extrapolated intensity of 553 nT. The Dst data are found to be insufficient for providing usefully accurate esti-mates of a statistically theoretical upper limit for magnetic storm intensity. Secular change in storm intensities is noted, as is a need for improved estimates of pre-1957 magnetic storm intensities.