A geostatistical approach to contaminant source estimation is presented. The problem is to estimate the release history of a conservative solute given point concentration measurements at some time after the release. A Bayesian framework is followed to derive the best estimate and to quantify the estimation error. The relation between this approach and common regularization and interpolation schemes is discussed. The performance of the method is demonstrated for transport in a simple one-dimensional homogeneous medium, although the approach is directly applicable to transport in two- or three-dimensional domains. The methodology produces a best estimate of the release history and a confidence interval. Conditional realizations of the release history are generated that are useful in visualization and risk assessment. The performance of the method with sparse data and large measurement error is examined. Emphasis is placed on formulating the estimation method in a computationally efficient manner. The method does not require the inversion of matrices whose size depends on the grid size used to resolve the solute release history. The issue of model validation is addressed.