This talk reviews some of the most recent achievements in the saga of the axiomatisation of parallel composition, along with some classic results. I will focus on the recursion, relabelling and restriction free fragment of CCS and I will discuss the solutions to three problems that were open for many years. The first problem concerns the status of Bergstra and Klop's auxiliary operators left merge and communication merge in the finite axiomatisation of parallel composition modulo bisimiliarity: I argue that, under some natural assumptions, the addition of a single auxiliary binary operator to CCS does not yield a finite axiomatisation of bisimilarity. Then I will delineate the boundary between finite and non-finite axiomatisability of the congruences in van Glabbeek's linear time-branching time spectrum over CCS.
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