The hypercube, though a popular and versatile architecture, has a major drawback in that its size must be a power of two. In order to alleviate this drawback, Katseff [1988] defined theincomplete hypercube, which allows a hypercube-like architecture to be defined for any number of nodes. In this paper we generalize this definition and introduce the namecomposite hypercube. The main result of our work shows that these incomplete architectures can be used effectively and without the size penalty. In particular, we show how to efficiently implement Fully Normal Algorithms on composite hypercubes. Development of these types of algorithms on composite hypercubes allows us to efficiently execute several algorithms concurrently on a complete hypercube. We also show that many host architectures, such as binary trees, arrays and butterflies, can be optimally embedded into composite hypercubes. These results imply that algorithms originally designed for any such host can be optimally mapped to composite hypercubes. Finally, we show that composite hypercubes exhibit many graph theoretic properties that are common with complete hypercubes. We also present results on efficient representations of composite hypercubes within a complete hypercube. These results are crucial in task allocation and job scheduling problems.