The association between an instance and a class is exclusive and permanent in many class-based, object-oriented database systems. Therefore, these systems have serious difficulties for applications in which objects takes on different and multiple roles over time. Recently, some researchers have proposed the use of roles to tackle these problems. Although many of them have realized the importance and difficulties to solve the type problems caused by roles, formal semantics and type issues of roles have not been addressed.In this paper, we develop an expressive yet semantically sound type calculus for objects with multiple roles. While the rich role modeling constructs are introduced, the calculus can still be expressed in a neat way, by using the polymorphism of overloaded functions. We show the important properties of the calculus which include Subject Reduction, Strong Normalization and Confluence. The calculus described in this paper provides a foundation for object-with-role models. We show that the calculus is general enough to be applied to various role models with little modification.