  We generalize the tensor product theory for modules for a vertex operator
algebra previously developed in a series of papers by the first two authors to
suitable module categories for a ``conformal vertex algebra'' or even more
generally, for a "M\"obius vertex algebra.'' We do not require the module
categories to be semisimple, and we accommodate modules with generalized weight
spaces. As in the earlier series of papers, our tensor product functors depend
on a complex variable, but in the present generality, the logarithm of the
complex variable is involved. This first part is devoted to the study of
logarithmic intertwining operators and their role in the construction of the
tensor product functors. Part II of this work will be devoted to the
construction of the appropriate natural associativity isomorphisms between
triple tensor product functors, to the proof of their fundamental properties,
and to the construction of the resulting braided tensor category structure.
This work includes the complete proofs in the present generality and can be
read independently of the earlier series of papers.
