Popular music is often composed of an accompaniment and a lead component, the latter typically consisting of vocals. Filtering such mixtures to extract one or both components has many applications, such as automatic karaoke and remixing. This particular case of source separation yields very specific challenges and opportunities, including the particular complexity of musical structures, but also relevant prior knowledge coming from acoustics, musicology or sound engineering. Due to both its importance in applications and its challenging difficulty, lead and accompaniment separation has been a popular topic in signal processing for decades. In this article, we provide a comprehensive review of this research topic, organizing the different approaches according to whether they are model-based or data-centered. For model-based methods, we organize them according to whether they concentrate on the lead signal, the accompaniment, or both. For data-centered approaches, we discuss the particular difficulty of obtaining data for learning lead separation systems, and then review recent approaches, notably those based on deep learning. Finally, we discuss the delicate problem of evaluating the quality of music separation through adequate metrics and present the results of the largest evaluation, to-date, of lead and accompaniment separation systems. In conjunction with the above, a comprehensive list of references is provided, along with relevant pointers to available implementations and repositories.