Steenrod homotopy theory is a natural framework for doing algebrate topology on general spaces in terms of algebraic topology of polyhendra, or from a different viewpoint, it studies the topology of the hm(1) functor (for inverse sequences of groups). This paper is primarily concenced with the case of compacta, in which Steenrod homotopy comcides with strong shape. An attempt is made to simplify the foundations of the theory and to clarify and improve some of its major results. With geometric tools such as Milnor's telescope compactification, comanifolds (=mock bundles), and the Pontryagin-Thom construction, new simple proofs are obtained for results by Barratt Milnoi, Geoghegan-Krasmkiewicz, Dydak, Dydak-Segal, Krasinkiewicz-Minc Cathey, Mittag-Leffler Bourbaki, Fox, Eda Kawamura, Edwards Geoghegan, Jussila, and the three unpublished results by Steenrod homotopy' is corrected it is shown that over compacta, R.H. Fox's overlayings are equivalent to I M James' uniforming coverning maps. Other results include. A morphism between inverse sequences of countable (possibly non-Abelian) groups that induces isomorphisms on lim and lim(1) is invertible the the pro-category. This implies the 'Whitehcad theorem in Steenord homotopy', thereby answering two questions of Koyama. If X is an LCn-1-compactum, n >= 1, then its n-dimensional Steenrod homotopy classes are representable by maps S-n -> X, provided that X is simply connected. The assumption of simple connectendness cannot be dropped, by a well-known results Dydak and Zdravkovska. A connected compatum is Steenrod connect (=pointed 1-movable), if and only if every uniform covering space of it has countably many uniform connected components. Bibliography: 117 titles
