This paper presents a general multi-camera visual-inertial navigation system (mc-VINS) with online instrinsic and extrinsic calibration, which is able to utilize all the information from an arbitrary number of asynchronous cameras. In particular, within the standard multi-state constraint Kalman Filter (MSCKF) framework, we only clone the IMU poses related to a single “base camera” (rather than all cameras) in the state vector, while the IMU poses corresponding to all other camera images are represented via an interpolation of the poses bounding the measuring time. By doing so, we can fuse all observations from all cameras with inertial measurements while allowing for efficient, tightly-coupled state estimation through parallelization and asynchrony. Moreover, we perform online sensor calibration of each camera's intrinsics as well as the spatial (transformation) and temporal (time offset) extrinsic parameters between all involved sensors (cameras and IMU), thus enabling high-fidelity localization. We validate the proposed mc-VINS algorithm in various real-world experiments with different sensor configurations, showing the ability to offer real-time high-precision localization and calibration results.