  Lattice paths effectively model phenomena in chemistry, physics and
probability theory. Asymptotic enumeration of lattice paths is linked with
entropy in the physical systems being modeled. Lattice paths restricted to
different regions of the plane are well suited to a functional equation
approach for exact and asymptotic enumeration. This thesis surveys results on
lattice paths under various restrictions, with an emphasis on lattice paths in
the quarter plane. For these paths, we develop an original systematic
combinatorial approach providing direct access to the exponential growth
factors of the asymptotic expressions.
