Electromagnetic field theory plays a very important role in present-day technology; examples of technologies based on electromagnetism that are inextricably bound up with every day life are: radar, remote sensing, geoelectromagnetics, bioelectromagnetics, antennas, wireless communication, optics, high-frequency circuits, and so on. In order to be able to develop new applications, and to improve the existing technologies, it is becoming increasingly important that there are methods available that accurately calculate electromagnetic fields. It is becoming particularly important to predict electromagnetic fields in inhomogeneous materials due to ongoing miniaturization of modern integrated circuits (ICs) and their ever-increasing operating frequencies. In the thesis we present the Volume-Integrated Field Equations (VIFE) method and the Surface-Integrated Field Equations (SIFE) method, two novel numerical methods for calculating time-harmonic electromagnetic fields in strongly inhomogeneous media. We calculate the electromagnetic fields with great accuracy for a number of complex 2D and 3D field problems. For instance, the thesis is concluded with the computation of the electromagnetic scattering of a dielectric cube where the domain of computation is truncated by using so-called ?Perfectly Matched Layers? (PMLs) that simulate extension to infinity.