TITLE: Vanishing cycles in a nutshell?
QUESTION [29 upvotes]: To quote one source among many, "the general reference for vanishing cycles is [SGA 7] XIII and XV". Is there a more direct way to learn the main principles of this theory (i.e. without the language of derived categories), in particular as it applies to the study of certain integral models of curves?

REPLY [16 votes]: Another good summary of vanishing cycles and Lefschetz pencils is found in Sections 4 and 5 of P. Deligne's "La conjecture de Weil, I" (available online). He discusses briefly both the theory over the complex numbers and the version in étale cohomology; and then immediately proceeds to putting these to pretty good use.