On single-valued and multi-valued convergences

We present a review of ideas of a general theory of convergence, developed independently of topology, with the stress on the duality of convergence and topology. Results and problems concerning sufficient and necessary conditions for a convergence to be topological, both in case of the single- and multi-valued cases, are recalled. We reconstruct, filling certain gaps, an example given in [7] to show that one of sufficient conditions in the theorems proved in [1] and [9] for multi-valued convergences to be topological is not necessary.