Maximal geodesics in lorentzian geometry The present thesis contains the first steps toward an Aubry Mather theory in the sense of V. Bangert D. Burago and J. Mather for Lorentzian manifolds. The first part studies the class of spacetimes (class A) to be considered in the subsequent Lorentzian Aubry Mather theory. Besides the necessary properties and objects (stabil timecone stabil time separation) a topological characterization of class A spacetimes is given analogous to the global splitting theorem of Geroch and Bernal Sanchez. The second part then devolopes the Lorentzian Aubry Mather theory for class A spacetimes. This encompasses the existence of maximal invariant measures and relations of the qualitive behavior of homologically maximizing geodesics to convexity properties of the subgraph of the stabil timecone. Additionally we study calibrations for class A spacetimes and prove the corresponding versions of Mathers graph theorem for maximal invariant measures. In the third part we define a subclass (class A_1) of class A spacetimes and prove the optimal multiplicity results as expected from the Riemannian case for maximal invariant measures in class A_1 spacetimes. The optimality is demonstrated with the Lorentzian Hedlund examples.