We describe the foundations of LamObj^a, a framework, or generic calculus, for modeling object-calculi. This framework is essentially a detailed formal operational semantics of object-based languages, in the style of the Lambda Calculus of Objects. As a formalism for specification LamObj^a is arranged in modules, permitting a natural classification of many object-based calculi according to their features. In particular there are modules for calculi of non-mutable objects (i.e., functional object-calculi) and for calculi of mutable objects (i.e., imperative object-calculi). As a computational formalism LamObj^a is based on rewriting rules. Classical first-order term rewriting systems are not appropriate since we want to reflect aspects of implementation practice such as sharing, cycles in data structures and mutation. Therefore we define the notion of addressed terms, and develop the corresponding notion of addresed term rewriting.