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The title of this paper is motivated by the title of the paper by Forsythewritten in 1952 and published in Bull. Amer. Math. Soc., 59 (1953), pp.299-329. Forsythe argues that solving a system of $n$ linear algebraicequations in $n$ unknowns is mathematically a lowly subject. His beautiful textgraduates with what was at that time ``the newest process on the roster, themethod of conjugate gradients.'' We consider it important to revisit, after 70years, to what extent Forsythe's views, and the views presented in the relatedcontemporary works of Hestenes, Stiefel, Lanczos, Karush, and Hayes, remainrelevant today.  Including, besides the conjugate gradient method (CG), also the generalizedminimal residual method (GMRES), we point out building blocks that we considercentral for the current mathematical and computational understanding of Krylovsubspace methods. We accomplish this through a set of computed examples. Wekeep technical details to a minimum and provide references to the literature.This allows us to demonstrate the mathematical beauty and intricacies of themethods, and to recall some persistent misunderstandings as well as importantopen problems. We hope that this work can initiate further theoreticalinvestigations of Krylov subspace methods.  This paper can not cover all Krylov subspace methods. The principlesdiscussed for CG and GMRES are, however, important for all of them. Further,practical computations always incorporate preconditioning. We will not dealwith preconditioning techniques, but we will deal with the basic question ofhow preconditioning is motivated, and we will recall some recent analyticresults.

Solving linear algebraic equations with Krylov subspace methods is still interesting!