We study the zero-temperature transport properties of one-dimensional normal metal-superconductor (NS) junctions with topological superconductors across their topological transitions. Working within the Blonder-Tinkham-Klapwijk (BTK) formalism generalized for topological NS junctions, we analytically calculate the differential conductance for tunneling into two models of a topological superconductor: a spinless intrinsic p-wave superconductor and a spin-orbit-coupled s-wave superconductor in a Zeeman field. In both cases we verify that the zero-bias conductance is robustly quantized at 2e(2)/h in the topological regime, while it takes nonuniversal values in the nontopological phase. The conductance spectra in the topological state develops a peak at zero bias for certain parameter regimes, with the peak width controlled by the strength of spin-orbit coupling and barrier transparency.