Usually, semantics of inheritance networks is specified indirectly through a translation into one of the standard logical formalisms. Since such translation involves an algorithmic aspect, which is usually complex, these approaches to inheritance are not truly declarative. We provide a general framework for specifying a direct semantics of inheritance networks. Because the networks are not expressive enough to capture all intuitions behind inheritance, a number of significantly different semantics have been proposed. Our approach allows us to give direct semantics to a number of different proposals found in the literature, and clarifies the relationships among them. It also provides a yardstick for measuring adequacy of translation into logical formalisms of various intuitions about inheritance.