In a topological superconductor besides the bulk superconducting gap that separates the normal and superconducting phases, there exist gapless edge states carrying spontaneous currents along the boundaries of the sample. Theoretical works have classified the topological superconductors as two types according to whether or not they break the time-reversal symmetry (the theoretical symmetry of physical laws under the transformation of time reversal).

Those are (i) chiral superconductors and (ii) helical superconductors, respectively. In superconductors bound pairs of electrons, namely Cooper pairs, are responsible for the emergence of superconductivity. In a chiral superconductor the Cooper pairs are spin polarized, i.e., spinless, owing to the breaking of the time-reversal symmetry (TRS), and its edge states resemble those of the quantum Hall state. On the other hand, in a helical superconductor the Cooper pairs are in a spin-triplet state, i.e., spinful owing to the TRS, and its edge states resemble those of the quantum spin Hall state.

The archetypal example of a topological superconductor breaking the TRS is the chiral p-wave model of superconductivity. Vortex cores in chiral p-wave superconductors are expected to host zero-energy modes (the condensed-matter equivalent of Majorana fermions), that are predicted to be the key element for the future quantum computation. Then, the interest in chiral p-wave superconductivity to develop a technological application from its topological properties appears well justified.

In this thesis, we studied chiral p-wave superconductivity to reveal the novel superconducting configurations that emerge in mesoscopic samples where confinement is of importance. The approach used in this thesis comprises the phenomenological Ginzburg-Landau theory and the microscopic Bogoliubov-de Gennes formalism, solved self-consistently. We discussed the novel magnetic, electronic and electric properties of the emergent states in order to facilitate the identification of chiral p-wave superconductivity in a candidate material. These features, namely the magnetic profile, the density of states, and the voltage-current characteristic, can be compared with results from Hall probe microscopy, scanning tunneling microscopy, and resistance measurements.