The main goal of the talk shall be to present certain insight into the theory of Riemann surfaces and their symmetries. Motivation for these studies comes from the fact, that every symmetric compact Riemann surface can be seen as a smooth projective complex algebraic curve having a real form, whose number of connected components equals the number of ovals of a symmetry. Therefore, one can study real forms of complex algebraic curves by studying Riemann surfaces and their symmetries, where combinatorial approach based on the Riemann uniformization theorem and theory of non-euclidean crystallographic groups is possible. During the talk we present some results concerning both quantitative and qualitative studies of the topic, showing in particular the maximal possible numer of nonconjugate symmetries on a Riemann surface of given genus and giving results concerning the topological properties of the set of points fixed by a system of symmetries.