The story of complete local rings is an old one. For us: a Cohen ring (of residue characteristic p) is a complete local ring A with maximal ideal pA. Under the extra hypothesis of regularity, these are complete discrete valuation rings. The structural results are due to many people, including famous names like Hasse, Schmidt, Witt, Teichmuller, and Mac Lane, in the 1930s. If the residue field is imperfect, there are some extra difficulties, worked out by Teichmuller and Mac Lane. In the 1940s, Cohen extended the structure theory by removing the hypothesis of regularity. We elaborate on his theory, supplementing the structural results with an `embedding lemma'. Turning to model theory, we apply the embedding lemma to give a relative quantifier elimination, and to describe the complete theories. Finally, if there is time, I will explain how we apply this to extend Belair's transfer of NIP from residue field to valued field in henselian mixed characteristic valued fields. (This is work in progress with Franziska Jahnke)