Notes on Nash modification

The Nash blowing-up (or modification) of an algebraic variety X is a canonical process that produces a proper, birational morphism π : X' → X of varieties. It is expected that the singularities of $X'$ will be better than those of X. In the mid-1970's, it was proved that in characteristic zero, π is an isomorphism if and only if X is nonsingular, which is false in positive characteristic. The focus of this article is on several subsequent studies on this subject. Topics covered include: (a) the extension of the mentioned theorem to the case where X is normal, in any characteristic, (b) the introduction and study of Nash modifications of higher order, (c) the case where the variety X is toric, where more precise results can be obtained and (d) desingularization properties of the Nash process.