This paper describes a general continuous-time framework for visual-inertial simultaneous localization and mapping and calibration. We show how to use a spline parameterization that closely matches the torque-minimal motion of the sensor. Compared to traditional discrete-time solutions, the continuous-time formulation is particularly useful for solving problems with high-frame rate sensors and multiple unsynchronized devices. We demonstrate the applicability of the method for multi-sensor visual-inertial SLAM and calibration by accurately establishing the relative pose and internal parameters of multiple unsynchronized devices. We also show the advantages of the approach through evaluation and uniform treatment of both global and rolling shutter cameras within visual and visual-inertial SLAM systems.