In Chapter 0, we give a brief discussion of positive linear functionals on C^*-algebras. We discuss the relation between states and representations of C^*-algebras in section 0.1. Some properties of factorial states and primal ideals of C^*-algebras are considered in section 0.2. In Chapter 1, we determine the primal ideals in certain C^*-algebras. We study an antiliminal C^*-algebra considered by Vesterstro m [38] in section 1.1. A variant of a nonliminal postliminal C^*-algebra constructed by Kadison, Lance and Ringrose ([18] and [3]) is considered in section 1.2. Also, we study in section 1.3 a liminal C^*-algebra constructed by J. Dixmier [10]. Chapter 2 is concerned with tensor products and primal ideals of C^*-algebras. In Chapter 3, we try to answer the following question 'when can the pure state space overlineP(A) be written as a union of weak^*-closed simplicial faces of the quasi state space Q(A)?' In section 3.1, we define a condition which we call (*), in terms of equivalent pure states, and we prove that it is equivalent to the pure state space overlineP(A) being a union of weak^*-closed simplicial faces of the state space S(A), where A is a unital C^*-algebra. We prove that the latter condition is equivalent to the pure state space overlineP(A) being a union of weak^*-closed simplicial faces of the quasi state space Q(A). In section 3.2, we consider questions of stability for the condition (*). Some C^*-algebras whose irreducible representations are of finite dimension are studied in section 3.3. Their pure state spaces are determined, thus giving examples and counter examples in connection with condition (*). Finally, in section 3.4, we prove that the factorial state space overlineF(A) is a union of weak^*-closed simplicial faces of Q(A) if, and only if, A is abelian.