Let S be a compact hyperbolic Riemann surface of genus $${g \geq 2}$$ g ≥ 2 . We call a systole a shortest simple closed geodesic in S and denote by $${{\rm sys}(S)}$$ sys ( S ) its length. Let $${{\rm msys}(g)}$$ msys ( g ) be the maximal value that $${{\rm sys}(\cdot)}$$ sys ( · ) can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface S max a compact Riemann surface of genus g whose systole has length $${{\rm msys}(g)}$$ msys ( g ) . In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating $${{\rm msys}(\cdot)}$$ msys ( · ) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.