On some results and problems in the theory of bases

The survey touches on the following three themes: (1) the order of decay of Fourier coefficients with respect to some orthogonal systems; (2) the order of approximation of functions of a given class by partial sums of orthogonal series; (3) the lowest possible order of growth of degrees of polynomials, constituting a basis of $C(0,1)$. A discussion of the present state of some unsolved problem is included and some new ones are posed.