In this paper, we present a monocular Direct and Semi-dense SLAM (Simultaneous Localization And Mapping) system for rolling shutter cameras. In a rolling shutter camera, the pose is different for each row of each image, and this yields poor pose estimates and poor structure estimates when using a state-of-the-art semi-dense direct method designed for global shutter cameras. To address this issue in tracking, we model the smooth and continuous camera trajectory using a B-spline curve of degree k??1 for poses in the Lie algebra, se(3).We solve for the camera poses at each row-time by a direct optimisation of photometric error as a function of the control points of the spline. Likewise for mapping, we develop generalised epipolar geometry for the rolling shutter case and solve for point depths using photometric error. Although each of these issues has been previously tackled, to the best of our knowledge ours is the first full solution to monocular, direct (feature-less) SLAM. We benchmark our method for pose accuracy and map accuracy against the state-of-the-art semi-dense SLAM system, LSD-SLAM, demonstrating the improved efficacy of our approach when using rolling shutter cameras via synthetic sequences with known ground-truth and real sequences.