Break-junctions (BJs) enable a pair of atomic-sized electrodes to be created and the relative position between them to be controlled with sub-nanometer accuracy by mechanical means - a level of microscopic control that is not yet achievable by top-down fabrication. Locally, a BJ consists of a single-atom contact, an arrangement that is ideal not only to study various types of quantum point contacts, but also to investigate transport through an individual molecule that can bridge such a junction. In this topical review, we will provide a broad overview on the field of single-molecule electronics, in which BJs serve as the main tool of investigation. To correlate the molecular structure and transport properties to gain a fundamental understanding of the underlying transport mechanisms at the molecular scale, basic experiments that systematically cover all aspects of transport by rational chemical design and tailored experiments are needed. The variety of fascinating transport mechanisms and intrinsic molecular functionalities discovered in the past range from nonlinear transport over conductance switching to quantum interference effects observable even at room temperature. Beside discussing these results, we also look at novel directions and the most recent advances in molecular electronics investigating simultaneously electronic transport and also the mechanical and thermal properties of single-molecule junctions as well as the interaction between molecules and light. Finally, we will describe the requirements for a stepwise transition from fundamental BJ experiments towards technology-relevant architectures for future nanoelectronics applications based on ultimately-scaled molecular building blocks.