In recent years, DC grids have gained momentum due to the advantages they display over their AC counterparts. For instance, high voltage DC grids have been established as the most prominent solution in the underwater power transmission or interconnection of asynchronous systems. Consequently, the success of high voltage DC grids is expected to be replicated in the medium voltage domain. However, medium voltage DC grids have not been standardized yet, leaving the space for various research topics. To realize a flexible and resilient DC grid, dependable connections among its parts operating under different voltage levels must be ensured. To put it differently, the DC-DC converter represents a keystone of any DC grid, operating at an arbitrary voltage level. Owing to the outstanding flexibility, scalability, and availability, the modular multilevel converter established itself in a vast variety of applications. Hence, the possibilities it offers in the high/medium voltage domain deserve to be further inspected. This thesis is divided into two parts. The first part concerns the modular multilevel converter operating as the DC-AC converter. The presented modeling approach, implying the averaging of the converter equations on the branch level, provided the grounds for the control-related discussions. The control of terminal currents was elaborated in detail and supported by a thorough analysis and comparison of the methods employed to maintain the proper converter energy distribution. Despite the theoretically unlimited voltage scalability, the modular multilevel converter power extension through the current capacity boost represents a significant technical challenge being comprehensively addressed throughout this thesis. It was shown that by paralleling the converter branches, the current, and implicitly the power, capacity of the converter can be increased while requiring no hardware adaptations of the existing parts. Moreover, branch paralleling offers the possibility to improve the spectral content of voltage at either of the converter terminals. The second part of this thesis refers to the employment of the modular multilevel converter in the domain of the DC-DC conversion. A novel, high power, bidirectional, DC-DC converter, utilizing the Scott transformer connection to provide galvanic separation between two of its ports, was proposed. It features the possibility of interconnecting a bipolar DC grid of high, or medium, voltage level with another DC grid of an arbitrary voltage level. As the Scott transformer connection comprises two separate transformer units, the proposed topology ensures that the operation can be maintained even in case either of the bipolar grid DC feeders is lost. A set of minor modifications to the above-mentioned topology leads to the structure providing the means for unidirectional energy flow. Nevertheless, control principles for both of the proposed solutions were derived and confirmed through the set of detailed simulations. Additionally, the unprecedented shift of the Scott transformer connection operating frequency towards the medium frequency range was proposed. A comprehensive analysis of the so-called quasi two-level converter leg was provided in the last part of this thesis. Quasi two-level operating principles imply the sequential and smooth transition of the converter leg AC voltage from one polarity to another. However, the transition occurs in the discrete time steps, being referred to as