A generalization of the usual column-strict tableaux (equivalent to a construction of R. C. King) is presented as a natural combinatorial tool for the study of finite dimensional representations of GL n ( C). These objects are called rational tableaux since they play the same role for rational representations of GL n as ordinary tableaux do for polynomial representations. A generalization of Schensted's insertion algorithm is given for rational tableaux, and is used to count the. multiplicities of the irreducible GL n -modules in the tensor algebra of GL n . The problem of counting multiplicities when the kth tensor power gl n ⊗ k is decomposed into modules which are simultaneously irreducible with respect to GL n and the symmetric group S k is also considered. The existence of an insertion algorithm which describes this decomposition is proved. A generalization of border strip tableaux, in which both addition and deletion of border strips is allowed, is used to describe the characters associated with this decomposition. For large n, these generalized border strip tableaux have a simple structure which allows derivation of identities due to Hanlon and Stanley involving the (large n) decomposition of gl n ⊗ k .