We examine the observability properties of visual-inertial navigation systems, with an emphasis on self-calibration of the six degrees-of-freedom rigid body transform be- tween a camera and an inertial measurement unit (IMU). Our analysis depends on a differential geometric formulation of the calibration problem, and on an algebraic test for the ‘observability rank condition’, originally defined by Hermann and Krener. We demonstrate that self-calibration of the camera-IMU transform is possible, un- der a variety of conditions. In contrast with previous work, we show that, in the pres- ence of a known calibration target, both the local gravity vector and the IMU gyroscope and accelerometer biases are simultaneously observable (given sufficient excitation of the system). Further and more generally, we show that for a moving monocular camera and IMU, the absolute scene scale, gravity vector, and the IMU biases are all simulta- neously observable. This result implies that full self-calibration is possible, without the need for any prior knowledge about the environment in which the system is operating.