The paper defines the identification problem for Discrete Event Systems (DES) as the problem of inferring a Petri Net (PN) model using the observation of the events and the available output vectors, that correspond to the markings of the measurable places. Two cases are studied considering different levels of the system knowledge. In the first case the place and transition sets are assumed known. Hence, an integer linear programming problem is defined in order to determine a PN modelling the DES. In the second case the transition and place sets are assumed unknown and only an upper bound of the number of places is given. Hence, the identification problem is solved by an identification algorithm that observes in real time the occurred events and the corresponding output vectors. The integer linear programming problem is defined at each observation so that the PN can be recursively identified. Some results and examples characterize the identified PN systems and show the flexibility and simplicity of the proposed technique. Moreover, an application to the synthesis of supervisory control of PN systems via monitor places is proposed. (c) 2008 Elsevier Ltd. All rights reserved.