Commutative coherent rings form a standard class of rings which generate commutative Noetherian rings. The notion of completion with respect to a maximal ideal is also standard, but it seems not to have been well studied for coherent rings. The aim of this book is to study completions for coherent regular local rings and the derived functors of completion on their module categories. In particular, the theory of L-complete modules due to Greenlees and May should have an analogue and a major goal is to establish which parts of it work for these rings.