The theory and mechanics of a statistical smoothing algorithm for estimating stress fields based on observed data are described. Incorporated into the algorithm are tunable parameters which allow the user to adjust the smoothness and fidelity of the fitted stress field. In addition, the user has the option of using robustness weights, or to weight the predicted stress field by assigning quality ranks to each data point. Fitted stress field can be displayed as either a gridded map (axes of maximum, or minimum, horizontal stress are shown as short bars located at each point in an evenly spaced grid), or as a stress trajectory map. Examples of predicted stress fields for southern California and western Canada, based on borehole elongation data, are displayed as stresstrajectory maps. A predicted paleostress field for the Spanish Peaks intrusion complex in Colorado, based on vertical dike orientations, is displayed as a gridded map. The smoothing algorithm is not limited to the analysis of stress orientation data. Any data consisting of a set of undirected lines measured at discrete locations are appropriate for input; possible candidates include strain data, mineral or intersection lineations, and lineaments. ¿ American Geophysical Union 1990