Multipactor and microwave corona breakdown are two high frequency discharges occurring within microwave components for space applications as well as within particle accelerators. The appearance of these phenomena results in the exponential growth of the electron population triggered by the interaction of the environmental charged particles with the microwave component. The consequence of these discharges is an electron cloud build-up that, for space applications, loads the Radio Frequency (RF) cavities. This entails the increase of the system noise level and the return loss, generating undesirable harmonics and creating a local augmentation of the temperature that may destroy internal components, with the subsequent failure of the mission. Concerning the particle accelerators, the electron cloud induces the deflection and the deformation of the particle beam, reducing its beam-to beam collision efficiency and causing the malfunctioning of the accelerator. The lack of a software or a method to predict the multipactor power breakdown threshold of RF components leads the scientific community to deal with high-cost test campaigns. Although the detection of the radiated power produced by the multipactor current was proposed as a breakdown detection method, no absolute levels of radiation were available to compare with measurements and validate this technique to assess the breakdown onset. In this thesis, an analytical expression to quantify the radiated power spectrum produced by a multipactor discharge within a parallel-plate waveguide has been developed. As there must be a mechanism to limit the growth of the electron population and the radiated power spec- trum depends on the number of electrons, the multipactor steady-state is also analyzed in this thesis. As multi- pactor breakdown is assumed to be detected when the saturation in the amount of electrons has been reached, an analytical expression of the electron population in the steady-state has been proposed in this thesis. This could help to predict the levels of radiation that must be detected when the breakdown occurs, and could be compared with the measurements to establish a rigorous multipactor detection criterion. Another part of the thesis deals with the development of a globally convergent algorithm to calculate very efficiently a large amount of roots of the cross-product of Bessel functions. This algorithm can be employed in the calculation of the radiated power spectrum produced by a multipactor discharge within a coaxial waveguide using the appropriated Green function, which needs the evaluation of these roots. Regarding the corona breakdown, this thesis intends to show the critical dependency of the breakdown power threshold on the ionization rate model. The ionization rate is a gas parameter that provides the amount of electrons released per unit time due to the ionization produced in the gas molecules as function of the elec- tric field amplitude. Three models of the ionization rate found in the literature are considered in this thesis and the corona breakdown power thresholds predicted are compared. Noticeable differences between the results obtained with each model have been found in some cases, which are analyzed in this work. Finally, in line with this thesis, a new high power SMA-like coaxial connector (in mass and size) has been developed. This thesis shows the design of the “Power Sub-miniature” (PSM) connector, which is capable to withstand high input powers without the risk of corona or multipactor breakdown to occur. The PSM has been manufactured and tested showing excellent multipactor withstanding capabilities. This connector is, in fact, able to withstand, at least, up to 1500 W input power for a pulsed signal of 2% duty-cycle in P-band (438 MHz) as well as in L-band (1124 MHz), without showing any trace of multipactor discharge. This implies an improvement of 50% compared to other powerful connectors such as the TNC. Concerning corona breakdown, the PSM is able to withstand more than 80 W CW without showing any sign of glowing or noticeable discharge, what is a challenge for coaxial connectors of this size at these input powers.