Underwater environments pose significant challenges for visual simultaneous localization and mapping (SLAM) systems due to limited visibility, inadequate illumination, and sporadic loss of structural features in images. Addressing these challenges, this article introduces a novel, tightly coupled acoustic-visual-inertial SLAM approach, termed AQUA-SLAM, to fuse a Doppler velocity log (DVL), a stereo camera, and an inertial measurement unit (IMU) within a graph optimization framework. Moreover, we propose an efficient sensor calibration technique, encompassing the multisensor extrinsic calibration (among the DVL, camera, and IMU) and the DVL transducer misalignment calibration, with a fast linear approximation procedure for real-time online execution. The proposed methods are extensively evaluated in a tank environment with ground truth, and validated for offshore applications in the North Sea. The results demonstrate that our method surpasses current state-of-the-art underwater and visual-inertial SLAM systems in terms of localization accuracy and robustness. The proposed system will be made open-source for the community.