We define an algebraic framework based on non-safe Petri nets, which allows one to express operations such as iteration, parallel composition, and transition synchronisation. This leads to an algebra of process expressions, whose constants and operators directly correspond to those used in Petri nets, and so we are able to associate nets to process expressions compositionally. The semantics of composite nets is then used to guide the definition of a structured operational semantics of process expressions. The main result is that an expression and the corresponding net generate isomorphic transition systems. We finally discuss a partial order semantics of the two algebras developed in this paper.