Multiple sclerosis (MS) is a chronic inflammatory and degenerative disease of the central nervous system, characterized by the appearance of focal lesions in the white and gray matter that topographically correlate with an individual patient's neurological symptoms and signs. Magnetic resonance imaging (MRI) provides detailed in-vivo structural information, permitting the quantification and categorization of MS lesions that critically inform disease management. Traditionally, MS lesions have been manually annotated on 2D MRI slices, a process that is inefficient and prone to inter-/intra-observer errors. Recently, automated statistical imaging analysis techniques have been proposed to extract and segment MS lesions based on MRI voxel intensity. However, their effectiveness is limited by the heterogeneity of both MRI data acquisition techniques and the appearance of MS lesions. By learning complex lesion representations directly from images, deep learning techniques have achieved remarkable breakthroughs in the MS lesion segmentation task. Here, we provide a comprehensive review of state-of-the-art automatic statistical and deep-learning MS segmentation methods and discuss current and future clinical applications. Further, we review technical strategies, such as domain adaptation, to enhance MS lesion segmentation in real-world clinical settings.