We apply tensor methods to formulate theories of electron correlation in nonorthogonal basis sets, The resulting equations are manifestly invariant to nonorthogonal basis transformations, between functions spanning either the occupied or virtual subspaces of the one-particle Hilbert space. The tensor approach is readily employed in either first or second quantization, As examples, second-order Moller-Plesset perturbation theory, and coupled cluster theory with single and double substitutions, including noniterative triples, are recast using the tensor formalism. This gives equations which are invariant to larger classes of transformations than existing expressions. Procedures for truncating these equations are discussed.