This paper investigates the problem of vision and inertial data fusion. A sensor assembling constituted by one monocular camera, three orthogonal accelerometers and three orthogonal gyroscopes is considered. The first paper contribution is the analytical derivation of all the observable modes, i.e. all the physical quantities that can be determined by only using the information in the sensor data acquired during a short time interval. Specifically, the observable modes are the speed and attitude (roll and pitch angles), the absolute scale and the biases affecting the inertial measurements. This holds even in the case when the camera only observes a single point feature. The analytical derivation of the aforementioned observable modes is based on a non standard observability analysis, which fully accounts the system non linearities. The second contribution is the analytical derivation of closed-form solutions which analytically express all the aforementioned observable modes in terms of the visual and inertial measurements collected during a very short time interval. This allows introducing a very simple and powerful new method able to simultaneously estimate all the observable modes without the need of any initialization or a priori knowledge. Both the observability analysis and the derivation of the closed-form solutions are carried out in several different contexts, including the case of biased and unbiased inertial measurements, the case of a single and multiple features, and in presence and absence of gravity. In addition, in all these contexts, the minimum number of camera images necessary for the observability is derived. The performance of the proposed approach is evaluated via extensive Monte Carlo simulations and real experiments.