An open issue in data-driven dependency parsing is how to handle non-projective dependencies, which seem to be required by linguistically adequate representations, but which pose problems in parsing with respect to both accuracy and efficiency. Using data from five different languages, we evaluate an incremental deterministic parser that derives non-projective dependency structures in O(n2) time, supported by SVM classifiers for predicting the next parser action. The experiments show that unrestricted non-projective parsing gives a significant improvement in accuracy, compared to a strictly projective baseline, with up to 35% error reduction, leading to state-of-the-art results for the given data sets. Moreover, by restricting the class of permissible structures to limited degrees of non-projectivity, the parsing time can be reduced by up to 50% without a significant decrease in accuracy.