Liver vessel segmentation from routinely performed medical imaging is a useful tool for diagnosis, treatment planning and delivery, and prognosis evaluation for many diseases, particularly liver cancer. A precise representation of liver anatomy is crucial to define the extent of the disease and, when suitable, the consequent resective or ablative procedure, in order to guarantee a radical treatment without sacrificing an excessive volume of healthy liver. Once mainly performed manually, with notable cost in terms of time and human energies, vessel segmentation is currently realized through the application of artificial intelligence (AI), which has gained increased interest and development of the field. Many different AI-driven models adopted for this aim have been described and can be grouped into different categories: thresholding methods, edge- and region-based methods, model-based methods, and machine learning models. The latter includes neural network and deep learning models that now represent the principal algorithms exploited for vessel segmentation. The present narrative review describes how liver vessel segmentation can be realized through AI models, with a summary of model results in terms of accuracy, and an overview on the future progress of this topic.