The development of crystalline porous materials with high chemical stability is of paramount importance for their practical application. Here, we report the synthesis of polyarylether-based covalent organic frameworks (PAE-COFs) with high crystallinity, porosity and chemical stability, including towards water, owing to the inert nature of their polyarylether-based building blocks. The PAE-COFs are synthesized through nucleophilic aromatic substitution reactions between ortho-difluoro benzene and catechol building units, which form ether linkages. The resulting materials are shown to be stable against harsh chemical environments including boiling water, strong acids and bases, and oxidation and reduction conditions. Their stability surpasses the performance of other known crystalline porous materials such as zeolites, metal-organic frameworks and covalent organic frameworks. We also demonstrate the post-synthetic functionalization of these materials with carboxyl or amino functional groups. The functionalized PAE-COFs combine porosity, high stability and recyclability. A preliminary application of these materials is demonstrated with the removal of antibiotics from water over a wide pH range.