This is the first part in a series of papers in which we introduce and develop a natural,
general tensor category theory for suitable module categories for a vertex (operator)
algebra. This theory generalizes the tensor category 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 “Möbius
vertex algebra.” We do not require the module categories to be semisimple, and we …