Let \(X\) be a closed Riemann surface. If \(X\) is a 3-sheeted covering of the Riemann sphere, then it is called a trigonal Riemann surface. If the covering \(f:X \to \hat{\mathbb{C}}\) is cyclic regular, then \(X\) is called a cyclic trigonal Riemann surface. A trigonal Riemann surface is said to be real if it admits an anticonformal involution \(\sigma\) such that \(f\circ \sigma=c\circ f\), where \(c\) denotes the complex conjugation. In this paper it is supposed that \(X\) has genus \(g\geq 5\). Then a necessary and sufficient condition for \(X\) to be cyclic (non-cyclic) trigonal is given by means of Fuchsian groups. Also a necessary and sufficient condition for \(X\) to be real cyclic (non-cyclic) trigonal is given by means of NEC groups. Moreover the authors determine the possible species of real cyclic trigonal Riemann surfaces according to the parity of \(g\). For a real non-cyclic trigonal one, it is shown that a classical theorem of Harnack gives the only restriction to the species. At the same time the authors give an answer to the question raised by B. H. Gross and J. Harris concerning to the real locus of a curve \(X\) [Ann. Sci. Éc. Norm. Supér., IV. Sér. 14, 157–182 (1981; Zbl 0533.14011)]. In the last section of this paper the authors describe antipodal trigonal Riemann surfaces.