We investigate the evolution of both the occupied and unoccupied electronic structure in representative compounds of the infinitely adaptive superlattice series (Sb 2 ) m -Sb 2 Te 3 (m = 0-3) by means of angle-resolved photoemission spectroscopy and time-delayed two-photon photoemission, combined with first-principles band-structure calculations. We discover that the topological nature of the surface states and their spin texture are robust, with dispersions evolving from linear (Dirac-like) to parabolic (Rashba-like) along the series, as the materials evolve from semiconductors to semimetals. Our findings provide a promising strategy for engineering the topological states with the desired flexibility needed for realizing different quantum phenomena and spintronics applications. Three-dimensional topological insulators (TIs) are materials where the topological properties of the insulating bulk band structure guarantee the existence of metallic surface states [1,2]. These topological surface states (TSS) are characterized by a Dirac-like energy-momentum dispersion and by a helical spin-momentum texture [3,4] that protects them against backscattering and localization [5,6]. TIs are promising materials for spintronics and provide an exceptional playground to study novel physical phenomena [7,8]. Possible applications depend on the ability to control the electronic properties of the TSS. Common strategies for achieving such a control involve chemical substitution and adsorbate doping [9][10][11][12][13]. Alternatively, it has been proposed that multilayer heterostructures could be used to obtain artificial TIs with tunable properties defined by the stacking sequences of the constituent two-dimensional building blocks [14][15][16].In this Rapid Communication, we demonstrate the possibility of TSS band-structure engineering in the naturally occurring homologous series of topological superlattices (Sb 2 ) m -(Sb 2 Te 3 ) n composed of Sb 2 bilayers (BLs) and Sb 2 Te 3 quintuple layers (QLs) [17,18]. This series enables a systematic investigation of the evolution of topological properties as a function of the stacking sequence of the constituent building blocks. Our results show that the topological states are remarkably robust and that their dispersion can be tuned in the explored range (m = 0-3; n = 1), and in bulk Sb, with an unchanging strong Z 2 index ν 0 = 1.All (Sb 2 ) m -(Sb 2 Te 3 ) n compounds display layered crystal structures produced by ordered stacking of Sb 2 Te 3 QLs and antimony BLs along the c axis of the hexagonal unit cell. This provides for an "infinitely adaptive series" of * marco.grioni@epfl.ch distinct compounds between the end member compositions Sb 2 Te 3 (m = 0) and Sb (n = 0) [17][18][19][20] that are known to be a TI and a topological semimetal, respectively [21][22][23][24][25][26][27]. Early experiments assessed the topological nature of bulk Sb and Sb-Bi alloys [21,22]. By contrast, the experimental observation of the predicted TSS in Sb 2 Te 3 has been delayed by the intrinsic p doping ...