Completely integrable systems are remarkable evolution equations, the best known of which is probably the Korteweg-deVries equation. Their many "symmetries", or conserved quantities, often allow for a detailed and in-depth description of their solutions.

We will present a number of new results concerning the Ablowitz-Ladik equation (AL). This is a classical, completely integrable discretization of the nonlinear Schroedinger equation. We will contrast its properties with those of one of the most celebrated discrete integrable system, the Toda lattice, while also illustrating the varied nature of the tools and ideas involved in the theory of completely integrable systems: geometric, algebraic, and functional analytic, among others.