Classifications coarser than shape

About thirty years ago, in the time of an intensive study of the shape theory, several classifications of metric compacta coarser than shape were introduced. Two of them have been of a special interest: The quasi-equivalence (K. Borsuk, 1976) and S-equivalence (S. Mardešić, 1978). In the last decade a much deeper view into these relations has been achieved. In attempt to characterize or, at least, describe them in a category framework on purpose of easier operative studying, several new classifications and new "shape" theories occurred. The most interesting among them are the coarse and weak shape theory. This paper is intended to be an exhaustive survey of the new classifications and corresponding theories, their mutual relations and the most important results.