The development of crystalline porous materials (CPMs) with high chemical stability is of paramount importance for their practical uses. Here we report the synthesis of novel polyarylether covalent organic frameworks (PAE-COFs) with high crystallinity, porosity and exceptional chemical stability due to the inert nature of polyarylether building blocks. We demonstrate that these materials can be stable against harsh chemical environments involving boiling water, strong acids/bases, oxidation and reduction conditions, which exceed all known CPMs, including zeolites, metal-organic frameworks (MOFs) and COFs. Furthermore, we explore their advantages as an efficient platform for structural design and functional evolution. The functionalized PAE-COFs combine porosity, high stability, and recyclability, and deliver outstanding performance in the removal of antibiotics from water over a wide pH range.