We review recent progress in the characterization of the atomic and electronic structure of pristine and adsorbate-covered Bi₂Se₃(0001) using state-of-the-art analysis tools like surface X-ray diffraction, X-ray absorption fine structure, and spin- and momentum-resolved photoemission spectroscopy. On the basis of several examples we show how the topological surface state is modified by structural modifications. One example is the shift of the Dirac point of the pristine Bi₂Se₃(0001) surface induced by trace amounts of carbon and its concomitant expansion of the topmost inter-layer spacing. Dosing with atomic hydrogen leads to the complete removal of selenium within the top quintuple layer leaving behind a bismuth bilayer on seleniumterminated Bi₂Se₃(0001) thereby leading to the appearance of hole-like Rashba-split surface states. Gold deposited by molecular beam epitaxy substitutes bismuth within the near surface region involving an opening of a gap in the topological surface state.