The research of this Ph. D. thesis is in the field of Quantum Computation and Quantum Information. A key problem in this field is the fragile nature of quantum states. This be comes increasingly acute when the number of quantum bits (qubits) grows in order to perform large quantum computations. It has been proposed that geometric (Berry) phases may be a useful tool to overcome this problem, because of the inherent robustness of such phases to random noise. In the thesis we investigate geometric phases and quantum holonomies (matrix-valued geometric phases) in many-body quantum systems, and elucidate the relationship between these phases and the quantum correlations present in the systems. An overall goal of the project is to assess the feasibility of using geometric phases and quantum holonomies to build robust quantum gates, and investigate their behavior when the size of a quantum system grows, thereby gaining insights into large-scale quantum computation. In a first project we study the Uhlmann holonomy of quantum states for hydrogen-like atoms. We try to get into a physical interpretation of this geometric concept by analyzing its relation with quantum correlations in the system, as well as by comparing it with different types of geometric phases such as the standard pure state geometric phase, Wilczek-Zee holonomy, Lévay geometric phase and mixed-state geometric phases. In a second project we establish a unifying connection between the geometric phase and the geometric measure of entanglement in a generic many-body system, which provides a universal approach to the study of quantum critical phenomena. This approach can be tested experimentally in an interferometry setup, where the geometric measure of entanglement yields the visibility of the interference fringes, whereas the geometric phase describes the phase shifts. In a third project we propose a scheme to implement universal non-adiabatic holonomic quantum gates, which can be realized in novel nano-engineered systems such as quantum dots, molecular magnets, optical lattices and topological insulators. In a fourth project we propose an experimentally feasible approach based on “orange slice” shaped paths to realize non- Abelian geometric phases, which can be used particularly for geometric manipulation of qubits. Finally, we provide a physical setting for realizing non-Abelian off-diagonal geometric phases. The proposed setting can be implemented in a cyclic chain of four qubits with controllable nearest-neighbor interactions. Our proposal seems to be within reach in various nano-engineered systems and therefore opens up for first experimental test of the non-Abelian off-diagonal geometric phase.