The author studies Riemann surfaces with constant curvature \(-1\) and signature \((g,n)\) that is with genus \(g\) and \(n\) boundary components which here are simple closed geodesics or cusps. A systole of a surface is a shortest closed geodesic which is not a boundary component. The paper contains numerous results concerning the following two main problems: I: For a fixed signature \((g,n)\) search the surfaces with a systole of maximal length. II: For a fixed signature \((g,n)\) search the surfaces with the maximal number of systoles. For example new global and local maximal surfaces of small genus for problem I are presented. A general theory and examples are given for problem II. It is completely solved for genus 2 and an upper bound for the number of systoles of closed surfaces is proved. Here definitely is not enough space to mention all the ideas and results in this paper. You should read it!