  The purpose of the present paper is to lay the foundations for a systematic
study of tensor products of operator systems. After giving an axiomatic
definition of tensor products in this category, we examine in detail several
particular examples of tensor products, including a minimal, maximal, maximal
commuting, maximal injective and some asymmetric tensor products. We
characterize these tensor products in terms of their universal properties and
give descriptions of their positive cones. We also characterize the
corresponding tensor products of operator spaces induced by a certain canonical
inclusion of an operator space into an operator system. We examine notions of
nuclearity for our tensor products which, on the category of C*-algebras,
reduce to the classical notion. We exhibit an operator system S which is not
completely order isomorphic to a C*-algebra yet has the property that for every
C*-algebra A, the minimal and maximal tensor product of S and A are equal.
