The advent and subsequent advances of semiconductor nanostructures are reviewed and their impacts on electronics and physics are discussed with focus on their relation with molecular beam epitaxy. We examine first quantum wells, superlattices and related structures and discuss how the quantum control of electrons in such layered structures has been perfected by manipulating the nucleation and impurity incorporation processes during the epitaxial growth. We then describe various epitaxial approaches to synthesize 10 nm scale quantum dot and quantum wire structures so as to offer new and desirable properties and functions and discuss how zero and one-dimensional electrons are controlled by the clever use of self-assembly, facet growth, and other novel growth schemes of nanostructures. (C) 2003 Elsevier Science B.V. All rights reserved.