One of the fundamental properties of the impulsive systems is analyzed: observability. Algebraic criteria for testing this property are obtained for the nonlinear case, considering continuous and discrete outputs. For this class of systems, observability is explored not only through time derivatives of the output, but also considering few discrete measurements at different time-instants. In this context, it is shown that nonlinear impulsive control systems can be strongly observable or observable over a finite time interval. A new rank condition based on the structure of the impulses is found to characterize observability of linear impulsive systems. It generalizes the celebrated Kalman criterion, for both kind of outputs, discrete and continuous. Finally, these results are tested and illustrated both on academic examples and on two impulsive dynamical models from biomedical engineering science.