The present paper is a survey of a part of the theory devoted to certain problems concerning the algebraic independence of the values of analytic functions, to quantitative results on estimates for the measure of transcendence or the measure of algebraic independence of numbers, to functional analogs of these results on the algebraic independence of solutions of algebraic differential equations, and estimates for the multiplicities of zeros for polynomials in these solutions, which play an important role in the proof of numerical results. This choice is related to the fact that, in December 2016, the head of the Department of Number Theory of Moscow State University, Corresponding Member of the RAS Yu.V. Nesterenko, who did a lot to develop these directions of the theory Transcendental numbers and whose works are marked by many awards, became seventy. He is a laureate of the Markov RAS Prize, 2006, of the Ostrovsky international prize, 1997, of the Hardy–Ramanujan Society Prize, 1997, and the Alexander von Humboldt Prize, 2003. Since the article is dedicated to the 70th anniversary of the birth of Yurii Valentinovich, we preface the scientific part with a brief biography.