In the present thesis, a problem almost as old as quantum physics itself is critically reviewed from a modern point of view. That problem is the so-called "quantum-phase problem", which was discovered as early as 1926 by Dirac and remains puzzling in some aspects to this day; its subject is a theoretical description of quantum phase, i.e. the phase of a quantized electromagnetic or other field described by harmonic oscillators. While in this thesis, this problem is not solved (it has indeed

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... hed a magnitude that by far eludes the scope of any reasonably-sized diploma thesis), a coherent and consolidated derivation of the two main operator-based phase theories developed to date, the Susskind/Glogower- and the Pegg/Barnett-formalism, is provided, by means of which key insights into the nature of phase and the unique difficulties plagueing its quantum description are gained. These insights include a systematic identification of the main issues at hand, an abstract reasoning about the existence of phase-operators and crosslinks to functional analysis and generalized measurement theory. These theoretical parts are then followed by a brief primer on the experiments that have been conducted so far, and by an overview over the topic of uncertainty and uncertainty relations in the specific context of quantum phase. We close with general observations and remarks. Throughout the thesis, special emphasis is put on providing the intermediate steps of most of the derivations instead of just reproducing the results given in the literature, a useful process which at once operates as a sanity-check and illustrates how to work with the quantities involved in any treatment of quantum phase.