Organ transplantation saves lives. Economic research based on mechanism design provides for optimal organ exchange when donors have certain incompatibility. Liver exchange has been practiced in small numbers, mainly to overcome blood-type incompatibility between patients and their living donors. A donor can donate either his or her smaller left lobe or the larger right lobe, although the former option is safer. We can think of liver exchange as a market-design model with simpler two-way exchanges, and introduce an individually rational, efficient and incentive-compatible mechanism for such a market. During the lecture, we will discuss how different technical tools from economic theory can not only help increase the number of transplants by more than 30%, but can also increase the share of the safer left-lobe transplants.