Modern hepatic surgery, and in particular the surgery of liver metastases, on patients with advanced and recurrent disease, as well as chemotherapy-induced liver injury, demands the pursuit of the apparently conflicting goals of radicality and tissue-sparing. Successful procedures require a perfect knowledge of the vascular anatomy of the liver, commonly based on Couinaud’s ideal representation that will be illustrated in detail. Alternative anatomical representations will be briefly presented, as they allow a better understanding of some surgical procedures such as central hepatectomies. We will argue that the best results will be obtained by deep understanding of the individual real anatomy of the patient, based on radiological reconstructions that are now more widely available on the surgeon’s laptop, and on intraoperative ultrasound. In addition, we will detail the anatomical characteristics of some structures of the liver, such as features particular to individual segments, the glissonean pedicles, the hepatic veins, the vestigial structures such as the umbilical and Arantius’ ligaments, and the surgical approaches and maneuvers that knowledge of these structures allows. The customized procedures that result go beyond the conventional segmental representation, are best described as tailored territorial liver resections, ft the concept of precision liver surgery to which the authors fully subscribe , and illustrate the evolution from surgical anatomy to anatomical surgery that was anticipated in earlier work.