Majorana zero modes are neutral zero-energy localized excitations emerging in the low-dimensional condensed matter systems, which are their own antiparticles. These excitations are topological with an intrinsic ground-state degeneracy, belonging to the (SU$_2$)$_2$ algebra and obeying the non-Abelian anyonic braiding statistics, which provides a possibility to implement the fault-tolerant topological quantum computing. As a result, enormous experimental efforts have focused on the realization of the Majorana zero modes, especially in the one-dimensional semiconductor-superconductor Majorana nanowires during the past decade. Although experiments have observed the zero-bias conductance peaks, these experimentally observed peaks are not robustly quantized as theoretically predicted for the signature of Majorana zero modes, and many other hallmarks of Majorana zero modes are yet to be unambiguously confirmed in experiments, which makes the experimentally observed zero-bias conductance peaks being interpreted as the Majorana zero modes questionable.

Therefore, in this dissertation, we carry out a detailed theoretical analysis of the experimental results, and classify the experimentally observed zero-bias conductance peaks into three types: the good (i.e., the actual topological Majorana zero modes), the bad (i.e., which is the partially-separated quasi-Majorana modes induced by the inhomogeneous potential and quantum dot in the nanowire), and the ugly (i.e., which is the trivial low energy fermionic state induced by random disorder). Our study concludes that almost all the current experimentally observed zero-bias conductance peaks in the publications are the ugly zero-bias conductance peaks, and future experiments should focus on the improvement of the material quality to reduce disorder.