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In the paper we construct and develop a fiber strong shape theory forarbitrary spaces over fixed metrizable space $\Bo$. Our approach is based onthe method of Marde\v{s}i\'{c}-Lisica and instead of resolutions, introduced byMarde\v{s}i\'{c}, their fiber preserving analogues are used. The fiber strongshape theory yields the classification of spaces over $\Bo$ which is coarserthan the classification of spaces over $\Bo$ induced by fiber homotopy theory,but is finer than the classification of spaces over $\Bo$ given by usual fibershape theory.

Fiber Strong Shape Theory for Topological Spaces