Abstract. 
This is an introductory review talk based on the recent joint
work of the speaker with F. Gestesy, K. Makarov, K. Zumbrun, M. Das,  A.
Sukhtaev and A. Pogan.

The Evans function is a Wronskian-type determinant used to detect isolated
eigenvalues of the differential operators obtained by linearizing partial
differential  equations about such special solutions as traveling waves.
This information is needed to decide if the traveling wave is stable or
unstable.

In a general setting, we define the Evans function utilizing exponential
dichotomies (a continuous time version of uniform hyperbolicity), the
Lyapunov exponents, and the Bohl spectrum (also known as the Sacker-Sell
spectrum). We will discuss how to compute the Evans function via a
(two-modified) Fredholm determinant of the Birman-Schwinger operator
pencil. This result is a generalization of a classical theorem of Jost and
Pais since the Evans function is known as the Jost function in scattering
theory. We will also relate the Evans function  to the classical
Weyl-Titchmarsh function for matrix Hamiltonian systems. An infinite
dimensional version of the Evans function will be considered as well; it is
used to study multidimensional problems on infinite cylinders. Finally, the
notion of the modified Jost solutions will be introduced to compute the
stability index, that is, the derivative of the Evans function at an
eigenvalue.