Observation problems in control systems literature generally refer to problems of estimation of state variables (or identification of model parameters) from two sources of information: dynamic models of systems consisting in first order differential equations relating all system quantities, and online measurements of some of these quantities. For nonlinear systems the classical approach stems from the work of R. E. Kalman on the distinguishability of state space points given the knowledge of time histories of the output and input. In the differential algebraic approach observability is rather viewed as the ability to recover trajectories. This approach turns out to be a particularly suitable language to describe observability and related questions as structural properties of control systems. The present paper is an update on the latter approach initiated in the late eighties and early nineties by J. F. Pommaret, M. Fliess, S. T. Glad and the author.