Steenrod homotopy theory is a framework for doing algebraic topology on
general spaces in terms of algebraic topology of polyhedra; from another
viewpoint, it studies the topology of the lim^1 functor (for inverse sequences
of groups). This paper is primarily concerned with the case of compacta, in
which Steenrod homotopy coincides with strong shape. We attempt to simplify
foundations of the theory and to clarify and improve some of its major results.