Existing extended target probability hypothesis density (ET-PHD) filters are insufficient in tracking weak extended targets. Hough transform-based track-before-detect methods are designed to detect the weak targets in a straight-line constant-velocity model. Therefore, this paper presents a novel method for detecting and tracking multiple maneuvering weak extended targets by a 3-dimensional Hough transform (3DHT) and multiple hypothesis tracking (MHT). The proposed method consists of two stages. In stage 1, the measurements in multiple scans are partitioned into overlapped time windows. The tracklets in each window can be detected by the 3DHT. In stage 2, the tracklets are associated to get the entire trajectories by the MHT. The tracklets of weak targets can be detected by the 3DHT in stage 1. Association in stage 2 is designed to detect maneuvering targets. Some false alarm tracklets could be built in stage 1. However, the false alarm tracklets are independent and unlikely to form a sequential trajectory in stage 2. Merely, the trajectories whose target likelihood ratio larger than a detection threshold can be confirmed as a target. Both the real data and the synthetic data are performed with the proposed approach and several existing algorithms. The result infers that the proposed approach is superior to the others with much less prior information that is necessary.