Recent years have seen a surge interest in realizing Majorana zero modes in condensed matter systems. Majorana zero modes are zero-energy quasiparticle excitations which are their own anti-particles. The topologically degenerate Hilbert space and non-Abelian statistics associated with Majorana zero modes renders them useful for realizing topological quantum computation. These Majorana zero modes can be found at the boundary of a topological superconductor. While preliminary evidence for Majorana zero modes in form of zero-bias conductance peaks have already been observed, confirmatory signatures of Majorana zero modes are still lacking.

In this thesis, we theoretically investigate the robustness of several signatures of Majorana zero modes, thereby suggesting improvement and directions that can be pursued for an unambiguous identification of the Majorana zero modes. We begin by studying analytically the differential conductance of the normal-metal--topological superconductor junction across the topological transition within the Blonder-Tinkham-Klapwijk formalism. We show that despite being quantized in the topological regime, the zero-bias conductance only develops as a peak in the conductance spectra for sufficiently small junction transparencies, or for small and large spin-orbit coupling strength. We proceed to investigate the signatures of Majorana zero modes in superconductor--normal-metal--superconductor junctions and show that the conductance quantization in this junction is not robust against increasing junction transparency. Finally, we propose a dynamical scheme to study the short-lived topological phases in ultracold systems by first preparing the systems in its long-lived non-topological phases and then driving it into the topological phases and back. We find that the excitations' momentum distributions exhibit Stuckelberg oscillations and Kibble-Zurek scaling characteristic of the topological quantum phase transition, thus provides a bulk probe for the topological phase.