Surveillance is a process that observes behaviour, recognises properties and identifies individuals. It has become a commonplace phenomenon in our everyday life. Many surveillance practices depend on the use of advanced technologies to collect, store and process data. We propose (i) an abstract definition of surveillance; and (ii) an abstract definition of identity, designed to capture the common structure of many disparate surveillance situations. We argue that the notion of identity is fundamental to surveillance. Rather than having a single identity, individuals have many identities, real and virtual, that are used in different aspects of their lives. Most aspects of life are subject to some form of surveillance, and observations and identities can be aggregated. The notion of identity needs to be theorised. Our analysis is very general and, at the same time, sufficiently precise to be the basis of mathematical models.