There is no useful universal definition of integrability, instead an
effective and usable definition depends on the class of equations that
will be considered. I will discuss three classes of equations and the
definition of integrability best suited for them.
They will be 1) Hamiltonian mechanics and Liouville integrability, 2)
Partial difference equations and the existence of multisoliton
solutions, and 3) lattice equations and multidimensional consistency.