We present a method for the automatic localization of facial landmarks that integrates nonrigid deformation with the ability to handle missing points. The algorithm generates sets of candidate locations from feature detectors and performs combinatorial search constrained by a flexible shape model. A key assumption of our approach is that for some landmarks there might not be an accurate candidate in the input set. This is tackled by detecting partial subsets of landmarks and inferring those that are missing, so that the probability of the flexible model is maximized. The ability of the model to work with incomplete information makes it possible to limit the number of candidates that need to be retained, drastically reducing the number of combinations to be tested with respect to the alternative of trying to always detect the complete set of landmarks. We demonstrate the accuracy of the proposed method in the face recognition grand challenge database, where we obtain average errors of approximately 3.5 mm when targeting 14 prominent facial landmarks. For the majority of these our method produces the most accurate results reported to date in this database. Handling of occlusions and surfaces with missing parts is demonstrated with tests on the Bosphorus database, where we achieve an overall error of 4.81 and 4.25 mm for data with and without occlusions, respectively. To investigate potential limits in the accuracy that could be reached, we also report experiments on a database of 144 facial scans acquired in the context of clinical research, with manual annotations performed by experts, where we obtain an overall error of 2.3 mm, with averages per landmark below 3.4 mm for all 14 targeted points and within 2 mm for half of them. The coordinates of automatically located landmarks are made available on-line.