  Fast matrix-by-matrix multiplication (hereafter MM) is a highly recognized
research subject. The record upper bound 3 of 1968 on the exponent of the
complexity MM decreased below 2.38 by 1987, applies to celebrated problems in
many areas of computing, and is extensively cited in the Theory of Computing.
Further decrease of the exponent remains a celebrated challenge. Acceleration
of MM in the Practice of Computing is a distinct challenge, because all known
algorithms supporting the exponents below 2.7733 improve straightforward MM
only for unfeasible MM of immense size, greatly exceeding the sizes of interest
nowadays and in any foreseeable future. We first survey the mainstream study of
the acceleration of MM of unbounded sizes, cover the progress in decreasing the
exponents of MM, comment on its impact on the theory and practice of computing,
and recall various fundamental concepts and techniques supporting fast MM and
naturally introduced in that study by 1980. Then we demonstrate how the curse
of recursion naturally entered the game of decreasing the record exponents.
Finally we cover the State of the Art of efficient feasible MM, including some
most efficient known techniques and algorithms as well as various issues of
numerical and symbolic implementation. We hope that our review will help
motivate and properly focus further effort in this highly important area.
