Abstract: The emergence of topological band theory in the last century ushered in a new era for condensed matter physics. In topological insulators and topological semimetals, these materials host a wide range of distinct surface and bulk topological electronic states that exhibit unique properties. Validating the existence of these bands and their associated physical phenomena relies on advanced characterization and measurement techniques, as well as extreme experimental environments such as ultralow temperature, ultrahigh pressure, and strong magnetic field. Characterization approaches include spectroscopic methods that directly observe the band structures, as well as transport studies investigating electrical and thermal behavior stemming from the topological quasiparticles and Berry curvature. Since the 1980s, numerous unique physical phenomena have been uncovered through these techniques, including quantum Hall effect, topological surface states, Dirac/Weyl fermion, negative magnetoresistance induced by chiral anomaly, giant anomalous Hall effect and spin-orbital polaron. The article introduces various advanced characterization techniques for topological materials, highlighting their unique charm in the field of condensed matter physics. By presenting the diverse forms and behaviors of topological electronic states, it promotes the rapid advancement of topological physics.