  The $\pi$ -calculus is used as a model for programminglanguages. Its contexts
exhibit arbitrary concurrency, makingthem very discriminating. This may prevent
validating desir-able behavioural equivalences in cases when more
disciplinedcontexts are expected.In this paper we focus on two such common
disciplines:sequentiality, meaning that at any time there is a single threadof
computation, and well-bracketing, meaning that calls toexternal services obey a
stack-like discipline. We formalise thedisciplines by means of type systems.
The main focus of thepaper is on studying the consequence of the disciplines
onbehavioural equivalence. We define and study labelled bisim-ilarities for
sequentiality and well-bracketing. These relationsare coarser than ordinary
bisimilarity. We prove that they aresound for the respective (contextual)
barbed equivalence, andalso complete under a certain technical condition.We
show the usefulness of our techniques on a number ofexamples, that have mainly
to do with the representation offunctions and store.
