  Gonchar-Stahl's $\rho^2$-theorem characterizes the rate of convergence of
best uniform (Chebyshev) rational approximations (with free poles) for one
basic class of analytic functions. The theorem itself, its modifications and
generalizations, methods involved in the proof and other related details
constitute an important subfield in the theory of rational approximations of
analytic functions and complex analysis.
  The paper briefly outlines essentials of the subfield. Fundamental
contributions by A. A. Gonchar and H. Stahl are in the center of the
exposition.
