This book is an exposition of some basic algebraic topology at the border between homology and homotopy. The emphasis is explained as follows. A Leverhulme Emeritus Fellowship was awarded to the first author for 2002-2004 to support making available, in one volume and in a consistent style, the work on crossed complexes and related higher homotopy groupoids carried out from 1974 to 2005 mainly by the first two authors. This work resulted in 12 joint papers as well as a number of other collaborations.