One of the defining characteristics of topological phases is the existence of edge and surface states at the boundaries of 2D and 3D topological materials. These edge states and surface states are topologically protected'', and evade Anderson localization. At the edge of a 2D topological insulator (TI), gapless edge states form a spin-momentum-locked 1D helical'' liquid. Rashba spin orbit coupling (RSOC) is often present as long as the inversion symmetry in the material is broken. RSOC enables unconventional impurity and inelastic electron-electron backscatterings. We derive the finite temperature conductivity correction from the most relevant inelastic interaction. At zero temperature, we show that the TI edge remains ballistic even with the combination of RSOC and non-magnetic impurities. In addition, strong inter-particle interactions are allowed for two proximate TI edges due to the RSOC. Such interactions are strictly forbidden for ideal quantum spin Hall insulator edges in the absence of RSOC. We identify a novel low-temperature transport regime which involves the interlocking of the two TI edges, which co-rotate as ``quantum gears.'' The Luttinger liquid parameter can be measured in the locking regime from a two-terminal conductance measurement. With respect to the 3D topological phases, we study the universal properties of the surfaces of the 3D topological superconductors (TSCs). We demonstrate the Chalker scaling and random matrix statistics in the surface states with weak and strong disorder. We also confirm numerically that the wavefunction multifractality and density of states in certain surface states follow the predictions of conformal field theory, rather than the strong disorder dominated Gade-Wegner fixed point. The multivalley surface state (spin or heat) transport is also investigated. We show that Altshuler-Aronov conductivity corrections always vanish on the multivalley TSC surfaces. We predict universal surface thermal and (if conserved) surface spin conductivities. The novel transport behaviors of the TIs and TSCs should provide new insights for characterizing new topological materials in the future experiments.