High harmonic generation with Laguerre Gaussian beams High harmonic generation is a versatile process for one thing useful to explore the structure of atoms or molecules during the generation itself and apart from that a source of bright short coherent extreme ultraviolet radiation. Thereby the harmonic radiation can be controlled by the shape of the driving laser with respect to its polarization or frequencies. Recent advances show that Laguerre Gaussian beams which carry in addition to their spin also orbital angular momentum can be utilized for high harmonic generation. In this thesis we analyze high harmonic generation with Laguerre Gaussian beams in the framework of the strong field approximation and show that this requires both the interaction of a single atom with the driving laser and the macroscopic superposition of all single atom contributions. We first investigate high harmonic generation with linearly polarized Laguerre Gaussian beams. There we show how the orbital angular momentum of the driving laser is transferred to the generated harmonics. Here we developed vivid photon diagrams to explain the conservation of orbital angular momentum. We then consider phase matching of the generated radiation in order to increase the conversion efficiency. In particular we analyze the coherence length at different positions in the generating beam. Furthermore we investigate high harmonic generation with a pair of counter rotating circularly polarized Laguerre Gaussian beams. Here we derive selection rules that take account of the conservation of energy spin and orbital angular momentum. In addition we show that the orbital angular momentum of the generated harmonics can be precisely controlled by the orbital angular momentum of the driving beam.