Undoubtedly, in theory of K3-surfaces the principal tool of study making the theory tractable is the global Torelli theorem (essentially stating that the isomorphism class of a surface is determined by that of its Hodge structure), together with the surjectivity of the period map (a description of the realizable Hodge structures). There are a few other classes of analytic varieties (most notably curves, from which the name originates, or Abelian surfaces) for which similar statements hold. I will try to discuss the version of the global Torelli theorem/surjectivity of the period map for cubic 4-folds in P5 (mostly due to Clair Voisin). Then, I will discuss a recent application of these statements to the classification of large configurations of 2-planes in cubic 4-folds.
(joint work with I. Itenberg and Ch.J. Ottem)