This monogram considers certain sequential and parallel algorithms on interval graphs, permutation graphs and trapezoid graphs. These graphs arise quite naturally in real-world applications. I have tried to present a rigorous and coherent theory. Proofs are constructive and are streamlined as much as possible. I have directed much attention to the algorithmic aspects of every problem. Algorithms are expressed in a manner that will make their adaption to a particular programming language relatively easy. The complexity of every algorithm is analyzed so that some measure of its efficiency can be determined. This monogram will be very useful for applied mathematicians and computer scientists at the research level. Many applications of the theoretical and computational aspects of the subject are described throughout the monogram. The topics covered in this monogram have been chosen to fill a vacuum in the literature, and their interrelation.