Approximation by rational functions in complex regions

Approximation by rational functions $R_n(z)$ (in the $C$ and $L_p$ metrics) on plane compacta is investigated. The possibility is studied of the coincidence of rational and polynomial approximations for all $n$, and some functions are described for which this coincidence holds. Approximations on finite sets of points are investigated, and an explanation is given of why there are functions which cannot be approximated by rational functions of degree not higher than $n$ (in the $C$ metric).