A computationally efficient approach to local learning with kernel methods is presented. The Fast Local Kernel Support VectorMachine (FaLK-SVM) trains a set of local SVMs on redundant neighbourhoods in the training set and an appropriate model for each query point is selected at testing time according to a proximity strategy. Supported by a recent result by Zakai and Ritov (2009) relating consistency and localizability, our approach achieves high classification accuracies by dividing the separation function in local optimisation problems that can be handled very efficiently from the computational viewpoint. The introduction of a fast local model selection further speeds-up the learning process. Learning and complexity bounds are derived for FaLK-SVM, and the empirical evaluation of the approach (with data sets up to 3 million points) showed that it is much faster and more accurate and scalable than state-of-the-art accurate and approximated SVM solvers at least for non high-dimensional data sets. More generally, we show that locality can be an important factor to sensibly speed-up learning approaches and kernel methods, differently from other recent techniques that tend to dismiss local information in order to improve scalability.