This thesis has been limited to real variable theory, except for one example in Laplace transform theory and in the treatment of the Residue Theorem where complex variable theory is necessary to construct the tool which is used in evaluating improper integrals in real variable theory. Many statements and theorems would necessarily have to be altered if the discussion were expanded to include all complex values. Improper integrals, as treated herein, are Riemann-type integrals. In the Riemann theory of integration the functions are assumed to be bounded, and the intervals or regions of integration are assumed to be bounded. Integrals are improper because the intervals or regions of integration are unbounded or because the functions are unbounded. The study of improper integrals is, therefore, an extension of the Riemann Theory which is taken as basic.