In this thesis, we consider the numerical solution of parabolic initial-boundary value problems with variable in space and time, possibly discontinuous coefficients. Such problems typically arise in the simulation of heat conduction problems, diffusion problems, but also for two-dimensional eddy current problems in electromagnetics. Discontinuous coefficients allow the treatment of moving interfaces like the rotation of an electrical motor. We recall two different approaches to prove that the continuous problem is well-posed in different settings (spaces) under quite general (physical) assumptions. In order to solve parabolic problems numerically, a vertical or horizontal method of lines is traditionally applied. However, in this thesis, an alternative approach is chosen. We treat time just as another variable and derive a conforming space-time finite element method. This introduces some challenges, but enables us to apply results from the existing and well investigated theory on elliptic boundary value problems. We show stability of the method, and, additionally, an a priori error estimate is provided. The case of local stabilizations, which are important for adaptivity, is also investigated. To study the method in practice, we introduced typical model problems in one, two, and three spatial dimensions. The implementation of our space-time finite element method is fully parallelized. The numerical studies were performed on the high performance computing cluster RADON1, and the outcomes verify the theoretical results.