This Research Note in Mathematics introduces a new Morse (index) theory that is easier to use, less technical and more flexible than existing theories for strongly indefinite functionals. It also contains an extensive survey of material that is now classical but has never appeared in book form. This text provides an accessible, self-contained treatment of the subject matter with full proofs of the major results, and complete with applications of the theory to Hamiltonian systems and other areas of current research, including wave equations, Chern-Simon functionals and Lorentzian geometry.