Discrete spatial symmetries affect ergodic transport through two-dimensional mesoscopic systems with chaos-inducing geometries and coherence lengths greater than the system's size. This thesis looks at those effects in symmetries both commuting and anticommuting with the current. Additionally the validity of the fractal Weyl law for quantum systems with mixed phase space is examined. Calculating the joint probability distribution of transmission eigenvalues of systems with discrete symmetries using random matrix theory shows that in the orthogonal universality class the transmission eigenvalues show repulsion of every second eigenvalue. For lead-preserving symmetries this behaviour is a result of superposition of independent sequences, while for symmetries mapping leads onto each other different one-point weights in the probability distribution prevent such an interpretation. This has the largest effect for narrow leads, as confirmed by numerical calculations. Microscopical understanding of symmetry effects is gained using a semiclassical approach to calculate the weak localization corrections and universal conductance fluctuations of systems where symmetry is broken by disturbing the internal symmetry of the system or displacing the leads. Duplicating random matrix theory results for perfect symmetry, this approach shows that symmetry effects vanish fast for small deviations from symmetry over a large part of the system while arbitrary deviations with a width smaller than the lead width have finite effect. Symmetry effects will show whenever leads overlap in part with mirrored leads. These results are confirmed with a crossover of random matrix ensembles. We examine the statistics of resonances in open systems with a mixed phase space. For this, we introduce a Husimi representation of decaying states based on the Schur decomposition of the time evolution operator. Applying this method to the open kicked rotator shows that a modified fractal Weyl law holds, which is cormected to emerging quantum-to-classical correspondence.