This thesis treats two general problem areas in the field of wave chaos.

The first problem area that we address concerns short wavelength tunneling

from a classically confined region in which the classical orbits are chaotic. We de-

velop a quantitative theory for the statistics of energy level splittings for symmetric

chaotic wells separated by a tunneling barrier. Our theory is based on the ran-

dom plane wave hypothesis. While the fluctuation statistics are very different for

chaotic and non-chaotic well dynamics, we show that the mean splittings of differ-

ently shaped wells, including integrable and chaotic wells, are the same if their well

areas and barrier parameters are the same. We also consider the case of tunneling

from a single well into a region with outgoing quantum waves.

Our second problem area concerns the statistical properties of the impedance

matrix (related to the scattering matrix) describing the input/output properties of

waves in cavities in which ray trajectories that are regular and chaotic coexist (i.e.,

`mixed' systems). The impedance can be written as a summation over eigenmodes

where the eigenmodes can typically be classified as either regular or chaotic. By

appropriate characterizations of regular and chaotic contributions, we obtain statis-

tical predictions for the impedance. We then test these predictions by comparison

with numerical calculations for a specific cavity shape, obtaining good agreement.