  The work is motivated by the papers [Ba1], [Ba2], [Ba7], [Ba11], [Be] and
[Be-Tu]. In particular, the strong homology groups of continuous maps were
defined and studied in [Be] and [Be-Tu]. To show that given groups are homology
type functor, it was required to construct the corresponding shape category. In
this paper, we study this very problem. In particular, using the methods
developed in [Ba7], [Ba-Ts]. the strong shape theory of continuous maps of
compact metric spaces, so-called strong fiber shape theory is constructed.
