A multiple hypothesis tracking approach to the segmentation of small 3D vessel structures is presented. By simultaneously tracking multiple hypothetical vessel trajectories, low contrast passages can be traversed, leading to an improved tracking performance in areas of low contrast. This work also contributes a novel mathematical vessel template model, with which an accurate vessel centerline extraction is obtained. The tracking is fast enough for interactive segmentation and can be combined with other segmentation techniques to form robust hybrid methods. This is demonstrated by segmenting both the liver arteries in CT angiography data, which is known to pose great challenges, and the coronary arteries in 32 CT cardiac angiography data sets in the Rotterdam Coronary Artery Algorithm Evaluation Framework, for which ground-truth centerlines are available.