  Engineering chiral $p$-wave superconductivity in semiconductor structures
offers fascinating ways to obtain and study Majorana modes in a condensed
matter context. Here, we theoretically investigate chiral $p$-wave
superconductivity in quantum dots and quantum rings. Using both analytical as
well as numerical methods, we calculate the quasiparticle excitation spectra in
these structures and the corresponding excitation amplitudes and charge
densities. In the topological regime, we can observe the chiral edge modes
localized at the boundaries and possessing finite energy in quantum dots and
quantum rings. By applying a magnetic field which is expelled from the quantum
ring, but which creates a flux that is an odd integer multiple of
$\Phi_0/2=\pi\hbar/e$, Majorana modes, that is, (approximately) degenerate edge
modes with zero energy and zero charge density, become possible in the
topological regime. Furthermore, we investigate finite-size effects that split
these degenerate edge modes as well as the effect of a magnetic field
penetrating into the superconducting region that can under certain
circumstances still support edge modes with approximately zero energy and
charge.
