  Motivated by classical results of Aubry, Davenport and Cassels, we define the
notion of a Euclidean quadratic form over a normed integral domain and an ADC
form over an integral domain. The aforementioned classical results generalize
to: Euclidean forms are ADC forms. We then initiate the study and
classification of these two classes of quadratic forms, especially over
discrete valuation rings and Hasse domains.
