  In this paper, we discuss several concepts of the modern theory of discrete
integrable systems, including:
  - Time discretization based on the notion of B\"acklund transformation;
  - Symplectic realizations of multi-Hamiltonian structures;
  - Interrelations between discrete 1D systems and lattice 2D systems;
  - Multi-dimensional consistency as integrability of discrete systems;
  - Interrelations between integrable systems of quad-equations and integrable
systems of Laplace type;
  - Pluri-Lagrangian structure as integrability of discrete variational
systems.
  All these concepts are illustrated by the discrete time Toda lattices and
their relativistic analogs.
