Inequalities that estimate the deviation of a value of a function at some point from its mean value using some characteristics of the function, are sometimes called Ostrowski-type inequalities. The first result of this kind was obtained by Ostrowski in 1938 and such inequalities were heavily studied since then. Ostrowski-type inequalities can be viewed as a partial case of the problem to find the deviation between operators, or as a simplest form of the problem of optimization of cubature formulae. At the same time, it appears that such inequalities are often a key step in solutions of other important extremal problems in approximation theory, including the problems of optimization of cubature formulae, the Stechkin problem about approximation of unbounded operators by bounded ones, inequalities for derivatives of Landau–Kolmogorov type and of Nagy type, and others. In this chapter we discuss a general approach to some extremal problems of approximation theory, which in particular allows us to obtain many Ostrowski-type inequalities for various classes of functions. We show how Ostrowski-type inequalities can be applied to obtain sharp inequalities for derivatives and related problems. Using Ostrowski-type inequalities as a primary tool, we obtain solutions for problems of optimization of cubature formulae.