  In this thesis we will study Feynman integrals from the perspective of
A-hypergeometric functions, a generalization of hypergeometric functions which
goes back to Gelfand, Kapranov, Zelevinsky (GKZ) and their collaborators. This
point of view was recently initiated by the works [74] and [150]. Inter alia,
we want to provide here a concise summary of the mathematical foundations of
A-hypergeometric theory in order to substantiate this viewpoint. This overview
will concern aspects of polytopal geometry, multivariate discriminants as well
as holonomic D-modules.
  As we will subsequently show, every scalar Feynman integral is an
A-hypergeometric function. Furthermore, all coefficients of the Laurent
expansion as appearing in dimensional and analytical regularization can be
expressed by A-hypergeometric functions as well. Moreover, we can derive an
explicit formula for series representations of each Feynman integrals, which is
in particular suitable for an algorithmic approach.
  In addition, the A-hypergeometric theory enables us to give a mathematically
rigorous description of the analytic structure of Feynman integrals (also known
as Landau variety) by means of principal A-determinants and A-discriminants.
This description of the singular locus will also comprise the various
second-type singularities. Furthermore, we will find contributions to the
singular locus occurring in higher loop diagrams, which seem to have been
overlooked in previous approaches. By means of the
Horn-Kapranov-parameterization we also provide a very efficient way to
determine parameterizations of Landau varieties. We furthermore present a new
approach to study the sheet structure of multivalued Feynman integrals by use
of coamoebas.
