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Modeling topology in medical image processing algorithms has emerged as a powerful technique for computing structural representations that are consistent with the underlying anatomy. When applied to high resolution images of the brain, these methods have proven to be extremely beneficial to neuroscientific studies in generating mathematical representations of the cerebral cortex and other brain structures, improving the analysis and visualization of functional activity, and allowing for group comparisons of brain geometry. Topological properties help model the global connectivity of structures without placing a bias on shape. In addition to providing anatomical consistency, topology-preserving algorithms also exhibit an improved robustness to noise. We provide an introduction to the main concepts in digital topology on which these algorithms are based and review their use in the segmentation of magnetic resonance (MR) brain images. Advances in the incorporation of topology constraints within medical image processing algorithms now allow computationally efficient segmentations that are consistent with the underlying anatomy. Modeling of topology complements commonly used models of local smoothness, such as Markov random fields, and statistical models of shape. We believe topological models will eventually reach similar levels of adoption into image processing algorithms as those other models.