The success of the supervised classification of remotely sensed images acquired over large geographical areas or at short time intervals strongly depends on the representativity of the samples used to train the classification algorithm and to define the model. When training samples are collected from an image or a spatial region that is different from the one used for mapping, spectral shifts between the two distributions are likely to make the model fail. Such shifts are generally due to differences in acquisition and atmospheric conditions or to changes in the nature of the object observed. To design classification methods that are robust to data set shifts, recent remote sensing literature has considered solutions based on domain adaptation (DA) approaches. Inspired by machine-learning literature, several DA methods have been proposed to solve specific problems in remote sensing data classification. This article provides a critical review of the recent advances in DA approaches for remote sensing and presents an overview of DA methods divided into four categories: 1) invariant feature selection, 2) representation matching, 3) adaptation of classifiers, and 4) selective sampling. We provide an overview of recent methodologies, examples of applications of the considered techniques to real remote sensing images characterized by very high spatial and spectral resolution as well as possible guidelines for the selection of the method to use in real application scenarios.