The study of topological phases has become one of the most active areas of the field of condensed matter physics. In the last decade, there has been immense experimental progress, owing to quantum leaps in materials science and the theoretical understanding of these systems. A remarkable example of topological phases is topological superconductors. These unusual superconductors are believed to host exotic Majorana quasiparticles whose underlying physics epitomizes the concept of ”spooky action at a distance” in quantum mechanics: Simply moving them around one another can change the occupation of fermionic modes. In fact, by braiding them in a complicated fashion, most of the gates necessary for building a quantum computer may be implemented. Furthermore, since quantum information is stored in non-local degrees of freedom, qubits based on Majoranas are believed to be inherently resilient to noise. Along with other promising future applications, this motivates the need for a better understanding of Majorana quasiparticles, which is the overarching theme of this PhD dissertation. Apart from providing a review of the relevant background, the dissertation provides new insights and methods regarding the physics of Majorana bound states. The original contributions in the thesis are contained in four projects. Project A tackles the problem of reading out the state of a Majorana box qubit, a minimal qubit where quantum information is stored in Majorana bound states. By using a novel Lindbladian approximation, we develop a Markovian theory of the dynamics of the reduced density matrix of a Majorana box qubit whose parity degrees of freedom has been converted to charge through the coupling of a quantum dot. This coupling splits the ground state degeneracy, and the dot is subjected to a fermion-number preserving interaction with environment modes during the readout. Our model contains the dynamics of the readout apparatus, and we find analytical expressions for the decay rates. These analytical expressions are easily applicable for a wide range of possible experiments, and we provide two experimentally relevant examples as case studies. In Project B, we calculate the dephasing dynamics of an isolated Majorana box qubit subjected to electromagnetic fluctuations in a capacitively coupled electric circuit. These fluctuations are treated as classically oscillating fields, allowing for an intuitive picture of the dephasing as accrued non-adiabatic corrections to the time evolution due to the shifting of the Majorana zero-energy mode. Since the exact form of the noise term is unknown, the corrections are calculated by statistically averaging over different noise functions by using the fluctuation-dissipation theorem. The problem of dephasing of Majorana qubits due to electromagnetic noise is refined in Project C, where we develop a model using a Bloch-Redfield approach, thus keeping the quantum mechanical nature of the environment modes. This allows us to go to low temperatures, and at zero temperatures we find a potential source of fidelity loss in Majorana parity readouts stemming from the fact that the zero-energy modes are dressed by the bosons, while the readout apparatus measures the bare Majorana modes. W e calculate this fidelity loss for a projective measurement of the bare Majoranas, and we find that this can lead to a considerable source of errors. In proposals using measurement-based braiding of Majorana qubits, this error source thus enters into all stages of initialization, manipulation and readout of the qubits.Finally, in project D we propose a model which generalizes Majorana zero-energy modes. It is constructed by starting from a bosonic model analogous to the Ising model, except where the local degreesof freedom and the global gauge symmetry are related to an arbitrary finite non-abelian group. By constructing a generalized Jordan-Wigner transformation, we map the model onto a local model with dyonic idegrees of freedom, meaning that they carry both a ”magnetic charge” in the form of a group element index, as well as an ”electric charge” corresponding to an irreducible representation of the group. We find that the model generically has a topological phase with zero-energy dyonic edge modes. The zero-energy modes are in general weak zero-energy modes, meaning that their degeneracy does not extend to all excited states. We discuss conditions under which the ground state is topologically ordered. When these conditions are not met, the ground states may be locally distinguishable, owing to the appearance of holographic symmetry operators localized on the boundary. The fusion rules of these dyonic zero-energymodes are discussed, but determining the braiding statistics remains an open problem. Together the four projects expand upon our knowledge of Majorana physics in the context of topological quantum computation as well as our general understanding of emergent anyonic zero-energy modes. Hopefully, they will serve to guide theoretical and experimental condensed matter research towards establishing the existence of non-abelian anyons as a scientific fact.