Geometric topology by crystallization theory: results and problems

Within geometric (or PL) topology, a representation theory exists, which makes use of a particular class of edge-coloured graphs - called crystallizations - to deal with PL-manifolds of arbitrary dimension, with or without boundary. The present paper is mainly devoted to review some recent developments of crystallization theory, and to show the existingrelationships with other "classical" representation methods for PL-manifolds, such as Heegaard splittings, branched coverings and surgery on framed links.