2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/224631This paper is based on talks I gave in Nagoya and Kinosaki in August of 2003. I survey, from my own perspective, Goodwillie's work on towers associated to continuous functors between topological model categories, and then include a discussion of applications to periodic homotopy as in my work and the work of Arone-Mahowald.This is the version published by Geometry & Topology Monographs on 29 January 2007Algebraic Topology55N20, 55P43, 55P47, 18G55text