Fossils are usually discovered broken or distorted, therefore reconstruction is inevitably the first step towards any comparative analysis. We outline a general methodological framework by which missing information about biological specimens can be estimated using geometric morphometric methods and discuss how this relates to effective paleoanthropological use of incomplete and distorted crania. Combining digital data resources with geometric morphometrics, we go beyond the assembly of fragments on the computer. As in a three-dimensional jigsaw puzzle, we first assemble the virtual pieces manually. Then we use landmarks, several hundred semilandmarks, and information from complete specimens to estimate missing coordinates and correct for distortion simultaneously. One can thus incorporate information from incomplete specimens in a comparative morphometric analysis while keeping track of the uncertainties that result from partial preservation or deformation. We exemplify our approach by reconstructing the fossil crania Arago XXI, Taung, and KNM-WT 15000. As different assumptions and algorithms lead to different estimations, there exists no "all-purpose" reconstruction. Instead one creates multiple reconstructions--a posterior distribution in a Bayesian sense. This distribution reflects uncertainty due to missing data values and sensitivity to prior assumptions. While there will typically be shape differences among equally plausible reconstructions, these different estimates might still support a single conclusion.