The original goal was the design and implementation of a proof verifier based on the very expressive formalism of set theory with a powerful inferential engine containing a wide collection of decision procedures (cf. [38]). Most of the research concentrated though on theoretical aspects (namely the decision problem for fragments of set theory), as shown by the numerous publications culminated in the monograph [22] which gives a complete account of the state-of-the-art of computable set theory up to 1989. In this paper we present some recent techniques and compare them with early approaches to the decision problem. In giving applications of such techniques, we sort of review the decision problem in set theory and related areas with respect to the classifications of the models one is looking for, namely pure set models with no individuals or stratified models, and with respect to the structure of the formulae considered, namely whether explicit quantification is allowed or not. It must be noted however that the present survey is far from being complete. For instance, decision procedures relative to weak axiomatizations of set theory (cf. [54, 55, 57, 62, 64, 65]) or in presence of the anti-foundation axiom are not surveyed (cf. [56, 61]). Also undecidability results are not reviewed here (cf. [19, 59, 63]). The paper is organized as follows. Section 2 introduces basic concepts about set hierarchies, models, realizations, and the decision problem.