Traditional feature matching methods, such as scale-invariant feature transform (SIFT), usually use image intensity or gradient information to detect and describe feature points; however, both intensity and gradient are sensitive to nonlinear radiation distortions (NRD). To solve this problem, this paper proposes a novel feature matching algorithm that is robust to large NRD. The proposed method is called radiation-variation insensitive feature transform (RIFT). There are three main contributions in RIFT. First, RIFT uses phase congruency (PC) instead of image intensity for feature point detection. RIFT considers both the number and repeatability of feature points and detects both corner points and edge points on the PC map. Second, RIFT originally proposes a maximum index map (MIM) for feature description. The MIM is constructed from the log-Gabor convolution sequence and is much more robust to NRD than traditional gradient map. Thus, RIFT not only largely improves the stability of feature detection but also overcomes the limitation of gradient information for feature description. Third, RIFT analyses the inherent influence of rotations on the values of the MIM and realises rotation invariance. We use six different types of multi-modal image datasets to evaluate RIFT, including optical-optical, infrared-optical, synthetic aperture radar (SAR)-optical, depth-optical, map-optical, and day-night datasets. Experimental results show that RIFT is superior to SIFT and SAR-SIFT on multi-modal images. To the best of our knowledge, RIFT is the first feature matching algorithm that can achieve good performance on all the abovementioned types of multi-modal images. The source code of RIFT and the multi-modal image datasets are publicly available1.