This paper develops a mathematical characterisation of object-oriented concepts by defining an observation-oriented semantics for an object-oriented language (OOL) with a rich variety of features including subtypes, visibility, inheritance, dynamic binding and polymorphism. The language is expressive enough for the specification of object-oriented designs and programs. We also propose a calculus based on this model to support both structural and behavioural refinement of object-oriented designs. We take the approach of the development of the design calculus based on the standard predicate logic in Hoare and He's Unifying Theories of Programming (UTP). We also consider object reference in terms of object identity as values and mutually dependent methods. © Springer-Verlag 2004.