There is a special classical case due to Lefschetz. In SGA 7.II, Deligne proposed the framework of vanishing and nearby cycle functors. Cf. also Milnor fiber.
Pierre Deligne, Le formalisme des cycles évanescents, Exposés XIII, XIV in:Groupes de monodromie en géométrie algébrique. II. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II). Dirigé par P. Deligne et N. Katz. Lecture Notes in Mathematics 340, Springer-Verlag 1973. x+438 pp, MR354657
wikipedia vanishing cycle; eom: vanishing cycle, monodromy transformation
S. Lefschetz, L’analysis situs et la géométrie algébrique, Gauthier-Villars, Paris, 1924; Jbuch 50, 663 (reprinted in his Selected papers, Chelsea Publ. 1971, MR299447)
V. Ginsburg, Characteristic varieties and vanishing cycles, Inv. Math. 84, 327–402 (1986)
Luc Illusie, Vanishing cycles over general bases after P. Deligne, O. Gabber, G. Laumon and F. Orgogozo, pdf
MathOverflow: is-there-a-good-way-to-think-of-vanishing-cycles-and-nearby-cycles, higher-vanishing-cycles
Chris Peters, Tata lectures on motivic aspects of Hodge theory, pdf
Masaki Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, pdf; in: Algebraic geometry (Tokyo/Kyoto, 1982), 134–142, Lecture Notes in Math. 1016, Springer 1983.
Mihai Tibăr, Polynomials and vanishing cycles, Cambridge Tracts in Mathematics 170, 2007. xii+253 pp. MR2008m:32048
Ryan Reich, Notes on Beilinson’s “How to glue perverse sheaves”, arxiv/1002.1686
Ofer Gabber, Gersten’s conjecture for some complexes of vanishing cycles, MR1305746, doi
There is a number of related preprints of D. Massey, cf. here including lecture notes on perverse sheaves and v.c. pdf
C. Sabbah, Proximité évanescente, Compos. Math. 62, 283-328, 1987.
Dirk Siersma, Vanishing cycles and special fibres, in: Singularity theory and its applications I_ (Coventry, 1988/1989), 292–301, Springer LNM 1462, 1991, MR92j:32129, doi
P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich, 2008. viii+326 pp. MR2009f:53143, doi
Sam Lichtenstein, Vanishing cycles for algebraic D-modules, Harvard senior thesis (adv. D. Gaitsgory) March 2009, pdf
A. Dimca, Sheaves in topology, Springer 2004, chap. 4.