Automatic segmentation of multiple sclerosis (MS) lesions in brain MRI has been widely investigated in recent years with the goal of helping MS diagnosis and patient follow-up. However, the performance of most of the algorithms still falls far below expert expectations. In this paper, we review the main approaches to automated MS lesion segmentation. The main features of the segmentation algorithms are analysed and the most recent important techniques are classified into different strategies according to their main principle, pointing out their strengths and weaknesses and suggesting new research directions. A qualitative and quantitative comparison of the results of the approaches analysed is also presented. Finally, possible future approaches to MS lesion segmentation are discussed.