  We give a comprehensive account of an analytic approach to spectral flow
along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or
$II_\infty$ von Neumann algebra ${\mathcal N}$. The framework is that of {\it
odd unbounded} $\theta$-{\it summable} {\it Breuer-Fredholm modules} for a
unital Banach *-algebra, $\mathcal A$. In the type $II_{\infty}$ case spectral
flow is real-valued, has no topological definition as an intersection number
and our formulae encompass all that is known. We borrow Ezra Getzler's idea
(suggested by I. M. Singer) of considering spectral flow (and eta invariants)
as the integral of a closed one-form on an affine space. Applications in both
the type I and type II cases include a general formula for the relative index
of two projections, representing truncated eta functions as integrals of one
forms and expressing spectral flow in terms of the JLO cocycle to give the
pairing of the $K$-homology and $K$-theory of $\mathcal A$.
