Systems with reduced numbers of leads that can synthesize the 12-lead electrocardiogram (ECG) with an insignificant or a small loss of diagnostic information have been proposed. The advantage over standard 12-lead ECG systems is the smaller number of measurement sites (i.e., electrodes) and, consequently, fewer wires. In this paper, we review all the important systems with reduced numbers of leads together with the methodology for synthesizing the leads. The fundamental theoretical background necessary to understand the most important concepts related to the synthesis is included. The presented theoretical and experimental justifications for the synthesis show that it is not necessary to measure a large number of leads directly, because the standard 12-lead ECG and arbitrary additional leads can be synthesized. Various approaches to evaluating the synthesized 12-lead ECG are defined and explained, and a number of systems that synthesize 12-lead ECG are presented as they were introduced in the literature. We cover the developments and improvements from the 1940s to the present day. The systems are classified on the basis of the synthesis method used, the approach to the evaluation of the synthesized ECG (depending on the measurement sites used), and on the number and types of leads employed. Based on a detailed assessment of state-of-the-art systems, open problems and challenges are highlighted, while further developments of electrocardiographic systems are envisaged.