  In this thesis we develop generalized versions of the Chung-Feller theorem
for lattice paths constrained in the half plane. The beautiful cycle method
which was developed by Devoretzky and Motzkin as a means to prove the ballot
problem is modified and applied to generalize the classical Chung-Feller
theorem. We use Lagrange inversion to derive the generalized formulas. For the
generating function proof we study various ways of decomposing lattice paths.
We also show some results related to equidistribution properties in terms of
Narayana and Catalan generating functions. We then develop generalized
Chung-Feller theorems for Motzkin and Schroeder paths. Finally we study
generalized paths and the analogue of the Chung-Feller theorem for them.
