Polysulfates and polysulfonates possess exceptional mechanical properties making them potentially valuable engineering polymers. However, they have been little explored due to a lack of reliable synthetic access. Here we report bifluoride salts (Q+[FHF]−, where Q+ represents a wide range of cations) as powerful catalysts for the sulfur(VI) fluoride exchange (SuFEx) reaction between aryl silyl ethers and aryl fluorosulfates (or alkyl sulfonyl fluorides). The bifluoride salts are significantly more active in catalysing the SuFEx reaction compared to organosuperbases, therefore enabling much lower catalyst-loading (down to 0.05 mol%). Using this chemistry, we are able to prepare polysulfates and polysulfonates with high molecular weight, narrow polydispersity and excellent functional group tolerance. The process is practical with regard to the reduced cost of catalyst, polymer purification and by-product recycling. We have also observed that the process is not sensitive to scale-up, which is essential for its future translation from laboratory research to industrial applications.