The main topics of this thesis are the geometric features of systolic complexesarising from the actions of hyperbolic isometries. The thesis consists ofan introduction followed by two articles.Given a hyperbolic isometry h of a systolic complex X, our central theme isto study the minimal displacement set of h and its relation to the actions of h onX and on the systolic boundary ∂X. We describe the coarse-geometric structureof the minimal displacement set and establish some of its properties that can beseen as a form of quasi-convexity. We apply our results to the study of geometricand algebraic-topological features of systolic groups. In addition, we provide newexamples of systolic groups.In the first article we show that the minimal displacement set of a hyperbolicisometry of a systolic complex is quasi-isometric to the product of a tree andthe real line. We use this theorem to construct a low-dimensional model for theclassifying space EG for a group G acting properly on a systolic complex, andto describe centralisers of hyperbolic isometries in systolic groups.In the second article we are interested in the induced action of h on thesystolic boundary, and particularly in the fixed points of this action. The maintheorem gives a characterisation of the isometries acting trivially on the boundaryin terms of their centralisers in systolic groups.