Three problem-dependent meanings for engineering “extremes” are motivated, established, and translated into formal geostatistical (model-based) criteria for designing infill sample networks. (1) Locate an area within the domain of interest where a specified threshold is exceeded, if such areas exist. (2) Locate the maximum value in the domain of interest. (3) Minimize the chance of areas where values are significantly different from predicted values. An example application on a simulated dataset demonstrates how such purposive design criteria might affect practice.