Computational aspects of crystallization theory: complexity, catalogues and classifications of 3-manifolds

The present paper is a survey of up-to-date results in 3-dimensional crystallization theory, in particular along the following directions:- generation and analysis of catalogues of PL-manifolds for increasing values of the vertex number of the representing graphs;- definition and/or computation of invariants for PL-manifolds, directly from the representing graphs.In particular, with regard to PL-manifold invariants, the authors focus on gems considered as an useful tool for computing Matveev complexity.