This book consists of a series of expository articles on axiomatic, enriched and motivic homotopy theory arising out of a NATO Advanced Study Institute of the same name at the Isaac Newton Institute for the Mathematical Sciences in Cambridge, UK in September 2002. There are introductions to abstract, stable and enriched homotopy theory, motivic homotopy categories, and stacks. Articles include applications to stable homotopy theory, higher commutativity, K-theory and descent. The list of authors brings together experts from both the homotopy theoretic and motivic sides, bridging the cultural divide between them.