Novel high-power medium-voltage converter technologies offering galvanic insulation are needed to support the development of the emerging medium-voltage direct-current grids and further improve the performance of various other applications such as traction, renewable energy and e-mobility. With the recent advancements in the power semiconductor industry, resulting in faster and more efficient switches with extended voltage and current capabilities, these converters, often referred to as solid state transformers (SSTs) or power electronic transformers (PETs), have become increasingly attractive. Besides the power semiconductor modules, the central component of any such converter, having a significant impact on its efficiency and power density, is the medium frequency transformer (MFT) that provides the necessary galvanic insulation and input-output voltage matching. However, the progress of the magnetic components has not been following the same pace as the semiconductor industry. Unlike the traditional line frequency transformers (LFTs), MFTs have not yet reached the technological maturity, thus leaving many areas open for research. To that end, this thesis focuses on the technical challenges tied to modeling and design optimization of the MFTs for the emerging SSTs. The available technologies and materials, suitable for medium frequency operation are identified and classified in respect to different application requirements. A detailed analysis and modeling of all the relevant phenomena governing the MFT electrical behavior as well as limiting the operation and design range is performed. A synthesis of all these models is done in form of a design optimization algorithm capable of generating the set of all feasible transformer designs. Moreover, design filters are developed allowing to interactively search for the most preferable design alternatives in terms of hot-spot temperatures, weight, volume and efficiency. As a proof of concept, a 100kW, 10kHz MFT prototype has been realized according to the optimal specifications resulting from the proposed design optimization tool. The accuracy of the utilized models was confirmed via thorough testing, including: electric parameter identification, partial-discharge test, full-power loading test and an extensive thermal run within the realized back-to-back resonant test setup. With the established design methodology and reliable models, a technology coordination and an MFT design sensitivity study has been performed on the SST level. A 0.5MW, 10kV input-series output-parallel series resonant converter based topology has been selected for the case study, taking into account different available semiconductor ratings and the resulting converter modularity. This study has shown that, considering the available technologies and materials, the expected MFT power density reaches its apex at around 10 −20kHz. While modern wide-band-gap semiconductors will for sure increase the efficiency and power density of the converter stage, this result indicates that further size reduction of the magnetic components above these frequencies will only be possible through improvements in the materials - providing core and winding materials with better high frequency loss characteristic.