  We define analytic torsion of Z_2-graded elliptic complexes as an element in
the graded determinant line of the cohomology of the complex, generalizing most
of the variants of Ray-Singer analytic torsion in the literature. It applies to
a myriad of new examples, including flat superconnection complexes, twisted
analytic and twisted holomorphic torsions, etc. The definition uses
pseudo-differential operators and residue traces. We also study properties of
analytic torsion for Z_2-graded elliptic complexes, including the behavior
under variation of the metric. For compact odd dimensional manifolds, the
analytic torsion is independent of the metric, whereas for even dimensional
manifolds, a relative version of the analytic torsion is independent of the
metric. Finally, the relation to topological field theories is studied.
