Artin Approximation

In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result and a similar result of P{\l}oski, he conjectured that this remains true when the ring of convergent power series is replaced by a more general kind of ring. This paper presents the state of the art on this problem, aimed at non-experts.