Topological superconductors with and without time-reversal symmetry are new phases of matters which host Majorana zero modes at their ends. The possibility of realizing such phases in various kinds of materials that are experimentally accessible, in addition to their unique signatures in simple transport measurements, has brought significant amount of attention from both theorists and experimentalists in condensed matter physics. In this thesis, we extend the previous studies on the realization of topological superconductors and try to answer some of the open questions regarding their transport signatures.

First, we study extensions of the realization scheme based on semiconducting nanowires proximity coupled to $s-$wave superconductors \cite{Lutchyn10, Oreg10} by replacing the s-wave superconductor with high temperature superconductors. We show that significant amount of induced superconducting gap in a nanowire can be achieve for a special interface geometry. The existence of gapless nodal excitations in the cuprate superconductors lead to a finite lifetime of Majorana zero modes when they are coupled to fermionic bath. We also consider the topological superconductivity in the Yu-Shiba-Rusinov states in chains of magnetic atoms at the surface of two dimensional $s-$wave superconductors with strong spin-orbit coupling. We study the generalization of the single Shiba state problem into a multiple Shiba states problem in the presence of spin-orbit coupling. We show that spin-orbit coupling induces the mixing of Shiba states correspond to different angular momentum channels and leads to interesting effects such as angular dependence of Shiba spectrum on the direction of magnetic moment. Based on these newly discovered effects, we propose new experimental methods to analyze and tune the physical parameters of the magnetic atom chains which can be applied to the ongoing experiments \cite{nadj2014, pawlak15, Ruby15}. Using the formalism developed for a single impurity, we study the magnetic atom chains with multiple Shiba state bands and present the topological phase diagram.

Next, we study the transport signatures of time-reversal invariant topological superconductors which support Kramers pair of Majorana modes. Especially, we explore the effects of interactions on the transport signatures in tunnel junctions involving Majorana Kramers pairs by considering two types of junction geometries. We first consider a junction between Majorana Kramers pair and Luttinger liquid. Using renormalization group (RG) analysis, we study the boundary conditions of the infrared fixed points where system flows to as a function of interaction strength. In the presence of weak repulsive interactions in the Luttinger liquid, two channel Andreev reflection is stable in contrast to the junction between an interacting lead and a conventional $s-$wave superconductor. Second, we study the ground state properties of Majorana Kramers pair-quantum dot-normal lead junction using weak coupling RG and slave-boson mean-field theory. We find that the Kodno interaction between the lead electrons and the quantum dot and the Majorana-quantum dot interaction compete each other. We find a new strong coupling fixed point characterized by strong correlation between impurity spin and Majorana Kramers pair, and we study its signatures in differential tunneling conductance.