Within the realm of finite permutation group theory there are a series of projects that could be collectively entitled The classification of finite combinatorial objects subject to transitivity assumptions. These kinds of classifications have, of course, been around a long time (for instance the Greeks interest in platonic solids is a particular instance) but the nature of this work changed very dramatically with the completion of the Classification of Finite Simple Groups.
Particular threads of this grand project include:
- The classification of distance-transitive graphs (cf. work of Saxl, Van Bon, Inglis and others);
- The classification of flag-transitive designs (cf. the paper of Buekenhout, Delandtsheer, Doyen, Kleidman, Libeck and Saxl which gives an almost-complete classification). More recently the flag-transitivity condition has been relaxed, and progress has been made on classifying designs which are, for instance, line-transitive or point-primitive (cf. work by many authors!)
- The classification of finite projective planes subject to various assumptions. This is a special case of the previous item. In 1959 Ostrom & Wagner gave a full classification of projective planes admitting 2-transitive automorphism groups; in 1987, and using CFSG, Kantor gave an almost-classification of projective planes admitting point-primitive automorphism groups; results have appeared subsequently dealing with the weaker situation of point-transitivity.
- The classification of generalized polygons subject to various assumption. The previous item is a special case of this. (I know of recent work on generalized quadrangles due to Bamberg, Giudici, Morris, Royle and Spiga; not sure about hexagons and octagons.)
- The classification of `special geometries'. This is work initiated (I believe) by Francis Buekenhout in an attempt to understand the sporadic groups (see the earlier answer by J Mckay). The idea is to find geometries on which the sporadic groups act, analogously to the way the groups of Lie type acts on Tits buildings.
- The classification of regular maps (i.e. graphs embedded nicely on topological surfaces and admitting an automorphism group that is regular on flags/ directed edges). This is the thread that involves the Platonic solids; more recently there is a wealth of work by people like Conder, Siran, Tucker, Jones, Singerman, and many others.
There are many others but these give a flavour (skewed to my own interests).
In many of the threads just mentioned (but not all) a crucial first step in classifying objects is to use the Aschbacher-O'Nan-Scott theorem which describes the maximal subgroups of $S_n$. One then often needs information about maximal subgroups of the almost simple groups and so another famous theorem of Aschbacher comes into play (along with results by Kleidman, Liebeck, and others). These theorems are closely related to the answer given by Gil Kalai - the production of results of this ilk (facts about the finite simple groups) is, in itself, a grand project!