This thesis is devoted to the topic of one- and two-dimensional models of topological superconductivity. We study the Kitaev chain subject to closed and open boundary conditions. In the closed chain, we derive the energy spectrum, the ground state, and the topological invariant viewed as a certain Berry phase. We study two-point correlation functions and find enlarged values close to the topological phase transitions. The open chain is studied with focus on describing the degenerate ground states for a simple parameter choice in the topological phase. Then, the open system Hamiltonian is diagonalized numerically, and we model the order parameter with a spatial dependency. Additional Majorana zero modes appear if the order parameter changes sign. We also consider the p+ip model. The localization of a Majorana zero mode, bound to a vortex that is described by Ginzburg-Landau theory, is found numerically. We propose an argument that results in a non-Abelian exchange transformation for a system of several vortices. The p+ip model is also studied on an annulus; we approximate the ground state as a combination of boundary states and calculate its energy. The results are compared to a numerical implementation, and the agreement is convincing as the boundary separation becomes large compared to other length scales.