higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
symmetric monoidal (∞,1)-category of spectra
A Grothendieck’s existence problem is a question of representability of a presheaf of sets (or sometimes of groups) on the category of affine schemes (or some variant of it) as a scheme (or as an algebraic space, Deligne-Mumford stack or some other algebraic variant of a space making the representability problem nontrivial…). In early days of scheme theory, at the end of 1950-s Grothendieck proved several major positive results of this sort, like the constructions of Hilbert, Quot and Picard schemes, and the Grothendieck’s existence theorem in formal geometry with major applications in deformation theory. Another major result of that sort is the Artin's representability theorem (and its refined version in higher geometry, the Artin-Lurie representability theorem).
Articles on the original theorem include
Luc Illusie, Grothendieck’s existence theorem in formal geometry, FGA explained, 179–233, MR2223409; (draft version pdf)
Michael Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira), Princeton Univ. Press, Princeton, N.J., 1969, pp. 21-71. MR0260746 (41:5369)
Siegmund Kosarew, Grothendieck’s existence theorem in analytic geometry and related results, Trans. Amer. Math. Soc. 328 (1991), 259-306, pdf
Discussion in E-∞ geometry is in