Semiconductor quantum dot molecules (QDMs) are systems composed of two or more closely spaced and interacting QDs. QDMs are receiving much attention both as playground for studying coupling and energy transfer processes between “artificial atoms” and as new systems, which substantially extend the range of possible applications of QDs. QDMs can be conveniently fabricated by self-assembly either through chemical synthesis or epitaxial growth. Although QDMs relying on the random occurrence of nearby QDs can be used for fundamental studies, special fabrication protocols must be used to create QDMs with well-defined properties. In this article, we focus on self-assembled QDMs obtained by epitaxial growth and embedded in a semiconductor matrix, which are appealing for the possible realization of quantum gates based on two-level systems defined in QDs. We provide a comprehensive overview of the development and current stage of the research on QDMs composed of vertically (in the growth direction) or laterally (in the growth plane) aligned QDs. The review highlights some recent milestone works and points out the challenges and future directions in the field.