Energy levels in the band gap arising from surface states can dominate the optical and electronic properties of semiconductor nanocrystal quantum dots (QDs). Recent theoretical work has predicted that such trap states in II-VI and III-V QDs arise only from two-coordinated anions on the QD surface, offering the hypothesis that Lewis acid (Z-type) ligands should be able to completely passivate these anionic trap states. In this work, we provide experimental support for this hypothesis by demonstrating that Z-type ligation is the primary cause of PL QY increase when passivating undercoordinated CdTe QDs with various metal salts. Optimized treatments with InCl3 or CdCl2 afford a near-unity (>90%) photoluminescence quantum yield (PL QY), whereas other metal halogen or carboxylate salts provide a smaller increase in PL QY as a result of weaker binding or steric repulsion. The addition of non-Lewis acidic ligands (amines, alkylammonium chlorides) systematically gives a much smaller but non-negligible increase in the PL QY. We discuss possible reasons for this result, which points toward a more complex and dynamic QD surface. Finally we show that Z-type metal halide ligand treatments also lead to a strong increase in the PL QY of CdSe, CdS, and InP QDs and can increase the efficiency of sintered CdTe solar cells. These results show that surface anions are the dominant source of trap states in II-VI and III-V QDs and that passivation with Lewis acidic Z-type ligands is a general strategy to fix those traps. Our work also provides a method to tune the PL QY of QD samples from nearly zero up to near-unity values, without the need to grow epitaxial shells.