  We develop a comprehensive theory of the stable representation categories of
several sequences of groups, including the classical and symmetric groups, and
their relation to the unstable categories. An important component of this
theory is an array of equivalences between the stable representation category
and various other categories, each of which has its own flavor (representation
theoretic, combinatorial, commutative algebraic, or categorical) and offers a
distinct perspective on the stable category. We use this theory to produce a
host of specific results, e.g., the construction of injective resolutions of
simple objects, duality between the orthogonal and symplectic theories, a
canonical derived auto-equivalence of the general linear theory, etc.
