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This is a draft version of a chapter for the Handbook of Logic and the Foundations of Computer Science, Oxford University Press. The final draft can be found as DAIMI PB 463. <br /> It surveys a range of models for parallel computation to include interleaving models like transition systems, synchronisation trees and languages (often called Hoare traces in this context), and models like Petri nets, asynchronous transition systems, event structures, pomsets and Mazurkiewicz traces where concurrency is represented more explicitly by a form of causal independence. The presentation is unified by casting the models in a category-theoretic framework. One aim is to use category theory to provide abstract characterisations of constructions like parallel composition valid throughout a range of different models and to provide formal means for translating between different models. It is still a draft at present. In particular, the ''Notes'' surveying related work are incomplete and the appendix on fibred categories needs to be overhauled in the light of some slick proofs, provided by Bart Jacobs. It is ragged in other places too. Constructive comments and corrections will be appreciated. <p>A knowledge of basic category theory is assumed, up to an acquaintance with the notion of adjunction.</p>

This is a survey of some decidability results for Petri nets, covering the last three decades. The presentation is structured around decidability o f speci c properties, v arious behavioural equivalences and nally the model checking problem for temporal logics.

An abstract definition of bisimulation is presented. It makes possible a uniform definition of bisimulation across a range of different models for parallel computation presented as categories. As examples, transition systems, synchronisation trees, transition systems with independence (an abstraction from Petri nets), and labelled event structures are considered. On transition systems the abstract definition readily specialises to Milner's strong bisimulation. On event structures it explains and leads to a strengthening of the history-preserving bisimulation of Rabinovitch and Traktenbrot and van Glabeek and Goltz. A tie-up with open maps in a (pre)topos, as they appear in the work of Joyal and Moerdijk, brings to light a new model, presheaves on categories of pomsets, into which the usual category of labelled event structures embeds fully and faithfully. As an indication of its promise, this new presheaf model has``refinement'' operators. The general approach yields a logic, generalising Hennessy Milner logic, which is characteristic for the generalised notion of bisimulation.

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