  We introduce and study the notion of a logarithmic vertex algebra, which is a
vertex algebra with logarithmic singularities in the operator product expansion
of quantum fields; thus providing a rigorous formulation of the algebraic
properties of quantum fields in logarithmic conformal field theory. We develop
a framework that allows many results about vertex algebras to be extended to
logarithmic vertex algebras, including in particular the Borcherds identity and
Kac Existence Theorem. Several examples are investigated in detail, and they
exhibit some unexpected new features that are peculiar to the logarithmic case.
