It is known since the works of Zariski in early 1940s that desingularization of varieties along valuations (called local uniformization of valuations) can be considered as the local part of the desingularization problem. This problem can be localized further to a problem of uniformizing transcendental extensions of valued fields, that is essentially a question from a (not so developed yet) theory of transcendental extensions of valued fields. In this lecture, I'll try to outline some parts of such a theory with accents on the following two theorems: generalized stability theorem, and inseparable uniformization of extensions of valued fields. An application to inseparable local uniformization will be briefly discussed (though it is not the central topic of the lecture).