The completion of a (normed) $C^*$ -algebra $\A_0[\| \cdot \|_0]$ with respect to a locally convex topology $\tau$ on $\A_0$ that makes the multiplication of $\A_0$ separately continuous is, in general, a quasi $*$ -algebra, and not a locally convex $*$ -algebra [10], [15]. In this way, one is led to consideration of locally convex quasi $C^*$ -algebras, which generalize $C^*$ -algebras in the context of quasi $*$ -algebras. Examples are given and the structure of these relatives of $C^*$ -algebras is investigated.