In this paper, we investigate the decidability and complexity of the fault diagnosis problem in unbounded labeled Petri nets. First, we show that checking diagnosability for unbounded Petri nets is decidable. We present a new necessary and sufficient condition for diagnosability, which can be reduced to a model checking problem for unbounded Petri nets. Then, we show that checking diagnosability for unbounded Petri nets is EXPSPACE-complete. This complexity result is further extended to various subclasses of Petri nets. To the best of our knowledge, this is the first paper that establishes decidability and complexity results for diagnosability of unbounded Petri nets.