This paper considers the problem of state estimation from a theoretical perspective. In particular, the investigation regards problems where the information provided by the sensor data is not sufficient to carry out the state estimation (i.e. the state is not observable). For these systems it is introduced the concept of continuous symmetry. Detecting the continuous symmetries of a given system has a very practical importance. It allows us to detect an observable state whose components are non linear functions of the original non observable state. This theoretical and very general concept is applied to deal with two distinct fundamental estimation problems in mobile robotics. The former is in the framework of self-calibration and the latter is in the framework of the fusion of the data provided by inertial sensors and vision sensors. For both problems all the observable modes are analytically derived by analyzing the continuous symmetries.