Abstract: Extensive growth in functional brain imaging, perfusion-weighted imaging, diffusion-weighted imaging, brain mapping and brain scanning techniques has led tremendously to the importance of the cerebral cortical segmentation, both in 2-D and 3-D, from volumetric brain magnetic resonance imaging data sets. Besides that, recent growth in deformable brain segmentation techniques in 2-D and 3-D has brought the engineering community, such as the areas of computer vision, image processing, pattern recognition and graphics, closer to the medical community, such as to neuro-surgeons, psychiatrists, oncologists, neuro-radiologists and internists. This paper is an attempt to review the state-of-the-art 2-D and 3-D cerebral cortical segmentation techniques from brain magnetic resonance imaging based on three main classes: region-based, boundary/surface-based and fusion of boundary/surface-based with region-based techniques. In the first class, region-based techniques, we demonstrated more than 18 different techniques for segmenting the cerebral cortex from brain slices acquired in orthogonal directions. In the second class, boundary/surface-based, we showed more than ten different techniques to segment the cerebral cortex from magnetic resonance brain volumes. Particular emphasis will be placed by presenting four state-of-the-art systems in the third class, based on the fusion of boundary/surface-based with region-based techniques outlined in Part II of the paper, also called regional-geometric deformation models, which take the paradigm of partial differential equations in the level set framework. We also discuss the pros and cons of various techniques, besides giving the mathematical foundations for each sub-class in the cortical taxonomy.