Digital libraries (DLs) are complex information systems and therefore demand formal foundations lest development efforts diverge and interoperability suffers. In this paper, we propose the fundamental abstractions of Streams, Structures, Spaces, Scenarios, and Societies (5S), which allow us to define digital libraries rigorously and usefully. Streams are sequences of arbitrary items used to describe both static and dynamic (e.g., video) content. Structures can be viewed as labeled directed graphs, which impose organization. Spaces are sets with operations on those sets that obey certain constraints. Scenarios consist of sequences of events or actions that modify states of a computation in order to accomplish a functional requirement. Societies are sets of entities and activities and the relationships between and among them. Together these abstractions provide a formal foundation to define, relate, and unify concepts - among others, of digital objects, metadata, collections, and services - required to formalize and elucidate "digital libraries". The applicability, versatility and unifying power of the 5S model are demonstrated through its use in three distinct applications: building and interpretation of a DL taxonomy, informal and formal analysis of case studies of digital libraries (NDLTD and OAI), and utilization as a formal basis for a DL description language.