Summary: In the paper, classical approximation theorems are investigated on the curves in a complex domain. In particular, direct and inverse theorems are cited on the curves \(\Gamma\) in a complex domain in the metric \(L_p(\Gamma)\). The obtained results remain new on the segment \([-1,1]\) as well. Furthermore, a new characteristic by which the authors obtain the analogies of Markov-Bernstein type and also S. M. Nikolskiy type classical estimates is also considered. In future, it is presupposed to get the analogies of classic theorems of Jackson, Jackson-Bernstein, Bernstein and Nikolskiy-Timan-Dzyadyk by these characteristics.