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We review the recently developed local spatial autocorrelation statistics Zi, Ci, Gi, and G,?. We discuss two alternative randomization assumptions, total and conditional, and then newly derive expectations and variances under conditional randomization for Zi and ci, as well as under total randomization for ci. The four statistics are tested by a biological simulation model from population genetics in which a population lives on a 21 x 21 lattice of stepping stones (sixtyyour individuals per stone) and reproduces and disperses over a number of generations. Some designs model global spatial autocorrelation, others spatially random sugaces. We find that spatially random designs give reliable test results by permutational methods of testing signif cance. Globally autocorrelated designs do not fit expectations by any of the three tests we employed. Asymptotic methods of testing signif came failed consistently, regardless of design. Because most biological data sets are autocorrelated, significance testing for local spatial autocorrelation is problematic. However, the statistics are informative when employed in an exploratory manner. We found that hotspots (positive local autocorrelation) and coldspots (negative local autocorrelation) are successfully distinguished in spatially autocorrelated, biologically plausible data sets. Work on spatial autocorrelation (SA) has been frequently featured in the pages of this journal. Pioneered in the 1950s, this topic was comprehensively treated by Cliff and Ord (1973, 1981). It was introduced to biology by Jumars, Thistle, and Jones (1977) and by Sokal and Oden (1978a, b). It has been extensively applied by ecologists and population biologists to a variety of organisms including humans, not only to test for departures from spatial randomness, but also to make inferences about the processes underlying the observed patterns. Representative studies are Sokal and Wartenberg (1981), Sokal and Jacquez (1991), Epperson (1993), Barbujani (1987), Sokal, Oden, and Barker (1987), and Sokal and Thomson (1998). Contribution no. 1016 in Ecology and Evolution from SUNY-Stony Brook. This research was supported by grant DEB9220538 from the National Science Foundation to Robert R. Sokal. The extensive computations were made possibIe by a grant of supercomputer time from the Cornell Theory Center.