Abstract Fast-Match is a fast algorithm for approximate template matching under 2D affifine transformations that minimizes the Sum-of-Absolute-Differences (SAD) error measure. There is a huge number of transformations to consider but we prove that they can be sampled using a density that depends on the smoothness of the image. For each potential transformation, we approximate the SAD error using a sublinear algorithm that randomly examines only a small number of pixels. We further accelerate the algorithm using a branch-and-bound scheme. As images are known to be piecewise smooth, the result is a practical affifine template matching algorithm with approximation guarantees, that takes a few seconds to run on a standard machine. We perform several experiments on three different datasets, and report very good results. To the best of our knowledge, this is the fifirst template matching algorithm which is guaranteed to handle arbitrary 2D affifine transformations.