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This paper is devoted to Markov's extremal problems of the form$M_{n,k}=\sup_{p\in\PP_n\setminus\{0\}}{{\|p^{(k)}\|}_X}/{{\|p\|}_X}$ $(1\lek\le n)$, where $\PP_n$ is the set of all algebraic polynomials of degree atmost $n$ and $X$ is a normed space, starting with original Markov's result inuniform norm on $X=C[-1,1]$ from the end of the 19th century. The central partis devoted to extremal problems on the space $X=L^2[(a,b);w]$ for the classicalweights $w$ on $(-1,1)$, $(0,+\infty)$ and $(-\infty,+\infty)$. Beside a shortaccount on basic properties of the (classical) orthogonal polynomials on thereal line, the explicit formulas for expressing $k$-th derivative of theclassical orthonormal polynomials in terms of the same polynomials arepresented, which are important in our study of this kind of extremal problems,using methods of linear algebra. Several results for all cases of the classicalweights, including algorithms for numerical computation of the best constants$M_{n,k}$, as well as their lower and upper bounds, asymptotic behaviour, etc.,are also given. Finally, some results on Markov's extremal problems on certainrestricted classes of polynomials are also mentioned.

On the Markov extremal problem in the $L^2$-norm with the classical weight functions