The Scale Invariant Feature Transform (SIFT) algorithm is widely used in computer vision to match features between images or to localize and recognize objets. However, mostly because of speckle noise, it does not perform well on synthetic aperture radar (SAR) images. We present here an improvement of this algorithm for SAR images, named SAR-SIFT. A new gradient computation, yielding an orientation and a magnitude robust to speckle noise, is first introduced. It is then used to adapt several steps of the SIFT algorithm to SAR images. We study the improvement brought by this new algorithm, compared to existing approaches. We present an application of SAR-SIFT for the registration of SAR images in different configurations, especially with different incidence angles.