This thesis studies the ω-completeness of process theories for parallel processes without communication, extended with constants for successful and unsuccessful termination. We look into the ω-completeness of the theories BSP,  which is the theory BSP extended with operators for parallel  composition and left merge, as well as PA0, which is the theory PA extended with a constant 0 for deadlock. While ground-completeness results are abundant in literature, previous work has failed to address the ω-completeness of process theories that are able to make the distinction between successful and unsuccessful termination of parallel processes. We will show that the theory BSP that we introduce is ω-complete by specifying behaviour on open terms alongside a generalised notion of bisimilarity ∼ on open terms and then show that the theory BSP is sound and complete with respect to ∼, from which the ω-completeness follows. We also attempt to show the ω-completeness of PA0, but run into a significant issue, which we leave as an open problem. Finally, we present an auxiliary result on the unique decomposition of PA0 terms into a parallel composition of parallel prime components.