Dual frequency capacitive discharges are designed to offer independent control of the flux and energy of ions impacting on an object immersed in the plasma. We investigate the operation of dual frequency discharges under a variety of geometries and operating conditions using, firstly, the electrostatic Particle-In-Cell (PIC) simulation method. We show that under certain conditions it is possible to obtain the desired independent control of both the flux and ion energy onto the electrodes. We find thought that within these discharges, the electron heating mechanisms are substantially different than their single frequency counterparts; under certain conditions the electron temperature becomes directly dependent on the voltage amplitude of the lower frequency power source. An analytical sheath model for a capacitively coupled radio-frequency plasma discharge operated with two frequencies is then proposed and studied under the assumptions of a time-independent, collisionless ion motion. Expressions are obtained for the time average electric potential within the sheath, nonlinear motion of the electron sheath boundary and nonlinear instantaneous sheath voltage. The derived model is valid under the condition that the low frequency (If) electric field E\f in the sheath is much higher than the high frequency (hf) electric field E^f. This condition is fulfilled within typical dual frequency conditions. It is shown, however, that the hf electric field modifies the sheath structure significantly as a result of the electron response to Ehf. This model has been compared to particle-in-cell plasma simulations, finding good quantitative agreement. We present the dependence of the maximum sheath width and the dc sheath voltage drop on the hf/lf current ratio and on the hf/lf frequency ratio. Subsequently, this analytical model is modified to describe the collisional ion dynamics within the sheath at higher more realistic pressure regimes. To describe the different ion dynamics, we have used a variable mobility model for the ion motion through the sheath. The sheath dynamics and characteristics of the collisionless and collisional models are then compared finding significant differences between the two models. A two dimensional PIC code is then developed to study the effect of operating plasma devices at greater frequencies than the normal industrial standard of 13.56 MHz. This PIC code is an Electromagnetic variant, meaning that the full set of Maxwells equations are solved for the fields, rather than simply Poissons equation. Using this PIC code it is found that the radial plasma density profile is increased significantly as the operating frequency is increased. This results in a greater uniformity of the ion bombardment profile onto the electrode.