A fundamental problem of the theory of quadratic forms is to classify quadrics over an arbitrary field up to irrational equivalence. For everywhere non-smooth quadrics in characteristic 2, we can solve some of the main problems about birational classification, which remain open for all other types of quadrics. As in Karpenko and Merkurjev’s theorem on the essential dimension of quadrics, the key is to study the Chow motives of quadrics (or, in more elementary terms, Chow groups of products of quadrics).