This book gives a systematic introduction to the index theory in nonlinear analysis. It presents fundamental methods and topics in the relative Morse index theories and their efficient applications to some boundary value problems of Hamiltonian systems and partial differential equations. For the relative Morse index theories, some new index theories which are suitable for investigating the non-periodic problems such as L-boundary value problems or P-boundary value problems of nonlinear Hamiltonian systems are presented. The main point of the topics in this book is to systematically deal with the non-periodic solution problems (open string problems) or the symmetric periodic solution problems of Hamiltonian systems