This collection consists of ten papers presented at the All-Union School on Function Theory, held in Dushanbe in August 1986, under the editor's guidance. The book encompasses a wide range of current directions in the metric theory of functions, the theory of approximation of functions, and related parts of mathematical analysis. The papers concern the following topics: extremal properties of functions, representation of functions by series, convergence of multiple Fourier series, approximation of functions by trigonometric polymonials in Lp-metrics, widths of classes of functions, approximation of functions by Fourier sums in systems of characters of zero-dimensional compact commutative groups, bilinear approximations of functions, the study of Tchebycheff sets in normed linear spaces, and spline approximation of functions of several variables. Among the results obtained are: new criteria for convexity of Tchebycheff sets in terms of continuity properties of the metric projection operator; conditions on the character of integrability of a periodic function of several variables under which its Fourier series converges to it in measure; a characterization of representation systems for symmetric spaces in which there are no nonzero continuous functionals.