Various versions of Artin approximation are widely used in Algebraic/Analytic Geometry, Commutative Algebra and Singularity Theory; traditionally, the approximation statements were restricted to Noetherian rings and to filtrations by powers of ideals. The goal of this talk is to explain how to extend some of the classical approximation results both for general filtrations and for rings of smooth functions; for instance, our statements allow some immediate applications of Artin approximation to the study of non-isolated singularities of maps and schemes. 
The content of this talk is based on joint work with Genrich Belitskii and Dmitry Kerner (Ben Gurion University of the Negev, Beer Sheva, Israel).