This book explores several aspects of discrete Hamiltonian systems. It is unique in that it provides interconnections between symplectic systems, recessive and dominant solutions, various discrete Riccati equations, and continued fraction representations of solutions of Riccati equations. It also presents variable step size discrete variational theory, a discrete Legendre transformation from discrete Euler-Lagrange equations to discrete Hamiltonian systems. Novel use is made of the implicit function theorem to show the importance of step size in numerical solutions of Hamiltonian systems. An `a priori' step size criterion shows how one can avoid parasitic numerical solutions. This book is accessible to students of mathematics, engineering, physics, chemistry and economics at the senior or beginning graduate level who have completed a course in matrix theory. It provides foundation work for engineering students studying optimal control and estimation as well as the variational problems arising in physics, chemistry, and economics. Audience: Mathematics, engineering and physics students who have had a course in matrix theory.