  We extend the definition of analytic and Reidemeister torsion from closed
compact Riemannian manifolds to compact Riemannian manifolds with boundary $(M,
\partial M)$, given a flat bundle $\Cal F$ of $\Cal A$-Hilbert modules of
finite type and a decomposition of the boundary $\partial M =\partial_- M \cup
\partial_+ M$ into disjoint components. In particular we extend the $L-2$
analytic and Reidemeister torsions to compact manifolds with boundary. If the
system $(M,\partial_-M, \partial_+M, \Cal F)$ is of determinant class we
compute the quotient of the analytic and the Reidemeister torsion and prove
glueing formulas for both of them. In particular we answer positively
Conjecture 7.6 in [LL]
