Treatment of atrial fibrillation (AF), the most common arrhythmia in clinical practice, remains challenging. Improved understanding of underlying mechanisms is needed to improve therapy. Functional re-entry is central to AF maintenance. The first detailed, quantitative theory of functional re-entry, the 'leading circle' model, was developed 40 years ago. Subsequently, an alternative paradigm based on 'spiral waves' has evolved. Spiral-wave generators, or 'rotors', have been identified using advanced mapping methods in experimental and clinical AF. A central tool in the analysis of spiral-wave rotors is the phase transformation, allowing for easier visualization of rotors and tracking of 'phase singularity' points at the rotor tip. In contrast to leading circle theory, which is expressed in terms familiar to (and easily understood by) cardiologists, the ideas needed to understand rotors are much more theoretical and harder for clinicians to apply. In this Review, we summarize the basic notions of phase mapping and spiral-wave rotors, and the ways in which rotor sources might be involved in AF maintenance. We discuss competing observations about the role of spatially confined rotors, short-lived rotors clustered at the edge of fibrotic zones, endocardial–epicardial interactive breeder properties and transmural re-entry, as well as studies underway to resolve them. We conclude with consideration of the clinical relevance of the issues discussed and their potential implications for the management of patients with AF.