  The complexity of matrix multiplication (hereafter MM) has been intensively
studied since 1969, when Strassen surprisingly decreased the exponent 3 in the
cubic cost of the straightforward classical MM to log 2 (7) $\approx$ 2.8074.
Applications to some fundamental problems of Linear Algebra and Computer
Science have been immediately recognized, but the researchers in Computer
Algebra keep discovering more and more applications even today, with no sign of
slowdown. We survey the unfinished history of decreasing the exponent towards
its information lower bound 2, recall some important techniques discovered in
this process and linked to other fields of computing, reveal sample surprising
applications to fast computation of the inner products of two vectors and
summation of integers, and discuss the curse of recursion, which separates the
progress in fast MM into its most acclaimed and purely theoretical part and
into valuable acceleration of MM of feasible sizes. Then, in the second part of
our paper, we cover fast MM in realistic symbolic computations and discuss
applications and implementation of fast exact matrix multiplication. We first
review how most of exact linear algebra can be reduced to matrix multiplication
over small finite fields. Then we highlight the differences in the design of
approximate and exact implementations of fast MM, taking into account nowadays
processor and memory hierarchies. In the concluding section we comment on
current perspectives of the study of fast MM.
