Opacity serves as a critical security and confidentiality property, which concerns whether an intruder can unveil a system’s secret based on structural knowledge and observed behaviors. Opacity in timed systems presents greater complexity compared to untimed systems, and it has been established that opacity for timed automata is undecidable. However, the original proof cannot be applied to decide the opacity of one-clock timed automata directly. In this paper, we explore three types of opacity within timed automata: language-based timed opacity, initial-location timed opacity, and current-location timed opacity. We begin by formalizing these concepts and establishing transformation relations among them. Subsequently, we demonstrate the undecidability of the opacity problem for one-clock timed automata. Furthermore, we offer a constructive proof for the conjecture regarding the decidability of opacity for timed automata in discrete-time semantics. Additionally, we present a sufficient condition and a necessary condition for the decidability of opacity in specific subclasses of timed automata.