  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 required; the general representation theory of vertex
operator algebras requires logarithmic structure. The first part of this work
is devoted to the study of logarithmic intertwining operators and their role in
the construction of the tensor product functors. The remainder of this work is
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.
