Emergent behaviour seems to be characteristic of the `complex systems’ that currently feature in Mathematics, Computer Science, the Physical Sciences and Philosophy. The purpose of this project is firstly to explore the nature of complexity and emergence in each of those areas by exploiting STIAS’s close connections. The Centre for Studies in Complexity is an important resource. Secondly it is to insist on quantitative reasoning about emergence. A model will be developed for expressing emergent behaviour using probability and nondeterminism, and for reasoning algebraically about it.