Simultaneous Localization and Mapping (SLAM) stands as one of the critical challenges in robot navigation. A SLAM system often consists of a front-end component for motion estimation and a back-end system for eliminating estimation drifts. Recent advancements suggest that data-driven methods are highly effective for front-end tasks, while geometry-based methods continue to be essential in the back-end processes. However, such a decoupled paradigm between the data-driven front-end and geometry-based back-end can lead to sub-optimal performance, consequently reducing the system's capabilities and generalization potential. To solve this problem, we proposed a novel self-supervised imperative learning framework, named imperative SLAM (iSLAM), which fosters reciprocal correction between the front-end and back-end, thus enhancing performance without necessitating any external supervision. Specifically, we formulate the SLAM problem as a bilevel optimization so that the front-end and back-end are bidirectionally connected. As a result, the front-end model can learn global geometric knowledge obtained through pose graph optimization by back-propagating the residuals from the back-end component. We showcase the effectiveness of this new framework through an application of stereo-inertial SLAM. The experiments show that the iSLAM training strategy achieves an accuracy improvement of 22% on average over a baseline model. To the best of our knowledge, iSLAM is the first SLAM system showing that the front-end and back-end components can mutually correct each other in a self-supervised manner.