The progress made during last years in the direction mentioned in the title is described. More precisely, `classical' theorems concerning stable and non-stable homotopic behavior of algebraic K-functors (starting point of which are Grothendieck-Serre's theorem on K_0-regularity of a regular ring and Quillen-Suslin's solution of Serre's problem) were generalized (including higher K-functors) to the monoid ring extensions corresponding to commutative cancellative monoids. In many cases these generalizations are stated in a final (i.e. maximal possible) form. However, recent minor results for higher K-functors show limitations in general of the posibility for such generalizations. Relevant ring theoretic topics are also treated.