Dielectric resonators (DRs) are being used extensively for microwave filters and oscillators because they offer several advantages such as small size, low cost, and, most important of all, freedom from metallic loss. It is shown that DRs of different shapes can also be used as effective radiators, namely the dielectric resonator antenna (DRA). Loss of the DR antenna is mainly caused by dielectric loss, which is very small in practice. These features make DRAs promising for high frequency applications.In this thesis, the hemispherical DRA is investigated. The hemispherical shape is chosen because a more accurate solution can be obtained without assumption of magnetic wall between dielectric and air. From the potential Green’s functions for an isolated spherical DRA the exact Green’s functions of various field components are derived easily. The Green’s functions are usually represented as the sums of a particular and a homogeneous solution in order to avoid the numerical difficulty encountered in evaluating higher-order wave functions. A physical argument is used to solve the singularity problem arising from the particular solution. In this thesis, the aperture coupled as well as the coaxial probe fed DR are investigated in Chapter 2 and Chapter 3, respectively. In the case of probe-fed excitation, a delta source is used for convenience.The fundamental problems considered in this thesis are the electromagnetic interactions of the fields due to impressed sources and a planar conducting screen having a slender slot cut in it. The screen is assumed to be perfectly conducting, vanishing thin, and of infinite extent. By equating the magnetic fields on the two sides of the slot interface, an integral equation for the slot electric field is established. We invoke image theory so that the derivations of the required Green’s functions are largely simplified.The unknown equivalent magnetic current along the slot can be determined by using the method of moments (MoM) with the Galerkin’s procedure. Both entiredomain sinusoidal basis (EB) functions and piecewise sinusoidal (PWS) basis functions are used when using the method of moments. Moreover, convergence checks for the MoM solutions are also performed. Once the magnetic current is found, the input impedance and the return loss etc. can be obtained readily. Partial theoretical results are compared to available experimental ones or measured ones in order to verify the theory.