Classes play a dual role in mainstream statically-typed object-oriented languages, serving as both object generators and object types. In such languages, inheritance implies subtyping. In contrast, the theoretical language community has viewed this linkage as a mistake and has focused on subtyping relationships determined by the structure of object types, without regard to their underlying implementations. In this paper, we explore why inheritance-based subtyping relations are useful and we describe two different approaches to extending the Moby programming language with inheritance-based subtyping relations. In addition, we present a typed object calculus that supports both structural and inheritance-based subtyping, and which provides a formal accounting of our extensions to Moby.