  This preprint appeared firstly in Russian in 1997. Some truncated versions of
this preprint were published in English and French, here a fully translated
version is presented. The translation in English was done by O. V. Feodoritova
and V. Deledicque to whom I express my gratitude.
  In the present paper I will describe how the first variant of the Godunov's
scheme has been elaborated in 1953-1954 and tell about all modifications
realized by myself (until 1969) and the group of scientists from the Institute
of Applied Mathematics in Moscow (which has become the M.V.Keldysh Institute of
Applied Mathematics).
