In his dissertation written at the University of Paderborn under the supervision of Prof. Dr. Joachim Hilgert, the author generalizes parts of a special non-Euclidean calculus of pseudodifferential operators, which was invented by S. Zelditch for hyperbolic surfaces, to symmetric spaces X=G/K of the noncompact type and their compact quotients spaces of nonpositive sectional curvature. Some results are restricted to the case of rank one symmetric spcaes. The non-Euclidean setting extends the defintion of so-called Patterson-Sullivan distributions, which were first defined by N. Anantharaman and S. Zelditch for hyperbolic systems, in a natural way to arbitrary symmetric spaces of the noncompact type. The author finds an explicit intertwining operator mapping Patterson-Sullivan distributions into Wigner distributions, he studies the important invariance and equivariance properties of these distributions and finally, he describes asymptotic properties of these distributions. Further research, results and generalizations will appear elsewhere in the future as a joint work together with J. Hilgert and S. Hansen.