Highly charged ions, in particular hydrogen-like ions, are simple systems which allow to observe a number of fundamental processes and quantities within physics at low energies. This comprises electromagnetic transitions in the fine and hyperfine structures, magnetic moments of bound electrons and ionic nuclei as well as the atomic mass of the electron and the value of the fine structure constant. In highly charged ions, the bound electrons are subjected to extreme fields which generate a number of interesting features. High precision is achieved by ion confinement in Penning traps where cooling of the ion motion and trap-specific manipulation and measurement techniques can be applied. This work reports on spectroscopy experiments for precision studies of highly charged ions in Penning traps and their connection to fundamental symmetries and constants.