This chapter is devoted to some well-known constructions of a function acting from R into R and nonmeasurable in the Lebesgue sense (respectively, of a function acting from R into R and lacking the Baire property). Obviously, the existence of such a function is equivalent to the existence of a subset of R nonmeasurable in the Lebesgue sense (respectively, of a subset of R without the Baire property). Since the fundamental concept of the Lebesgue measure on R (respectively, the concept of the Baire property) was introduced, it has been extremely useful in various problems of mathematical analysis. The natural question arose whether all subsets of R are measurable in the Lebesgue sense (respectively, whether all subsets of R possess the Baire property). Very soon, two essentially different constructions of extraordinary point sets in R were discovered which gave simultaneously negative answers to these two questions. The first construction is due to Vitali [272] and the second one was carried out by Bernstein [19]. Both of them were heavily based on an uncountable form of the Axiom of Choice, so it was reasonable to ask whether it is possible to construct a Lebesgue nonmeasurable subset of R (or a subset of R without the Baire property) by using some weak forms of the Axiom of Choice which are enough for most domains of classical mathematical analysis (for instance, the Axiom of Dependent Choices). Almost all outstanding mathematicians working at that time in mathematical analysis and particularly in the theory of real functions (Borel, Lebesgue, Hausdorff, Luzin, Sierpiński, etc.) believed that there is no effective construction of a Lebesgue nonmeasurable subset of R. However, only after long-term developments in mathematical logic and axiomatic set theory and, especially, after the creation (in 1963) of the forcing method by Cohen, did it become possible to establish the needed result. We shall return to this theme in our further considerations and touch upon some related problems that are also interesting from the logical point of view. But 154first, we wish to discuss more thoroughly analytic aspects of the problem of the existence of Lebesgue nonmeasurable point sets (respectively, of point sets without the Baire property).