The NATO complicated research Institute "Axiomatic, enriched and rna­ tivic homotopy idea" came about on the Isaac Newton Institute of Mathematical Sciences, Cambridge, England in the course of 9-20 September 2002. the administrators have been J.P.C.Greenlees and I.Zhukov; the opposite or­ ganizers have been P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The name describes the content material good, and either the development and the contents of the current quantity mirror contemporary impressive successes in version categor­ ies, based ring spectra and homotopy thought of algebraic geometry. The ASI took the shape of a sequence of 15 minicourses and some additional lectures, and was once designed to supply heritage, and to deliver the par­ ticipants modern with advancements. the current quantity is predicated on a few of the lectures given through the workshop. The ASI used to be the hole workshop of the 4 month programme "New Contexts for sturdy Homotopy thought" which explored numerous topics in higher intensity. i'm thankful to the Isaac Newton Institute for offering such a great venue, the NATO technology Committee for his or her investment, and to all of the audio system on the convention, whether they have been capable of give a contribution to the current quantity. All contributions have been refereed, and that i thank the authors and referees for his or her efforts to slot in with the tight agenda. eventually, i want to thank my coorganizers and the entire employees on the Institute for making the ASI run so easily. J.P.C.GREENLEES.