Microphysiology systems (MPS), also called organs-on-chips and tissue chips, are miniaturized functional units of organs constructed with multiple cell types under a variety of physical and biochemical environmental cues that complement animal models as part of a new paradigm of drug discovery and development. Biomimetic human liver MPS have evolved from simpler 2D cell models, spheroids and organoids to address the increasing need to understand patient-specific mechanisms of complex and rare diseases, the response to therapeutic treatments, and the absorption, distribution, metabolism, excretion and toxicity of potential therapeutics. The parallel development and application of transdisciplinary technologies, including microfluidic devices, bioprinting, engineered matrix materials, defined physiological and pathophysiological media, patient-derived primary cells, and pluripotent stem cells as well as synthetic biology to engineer cell genes and functions, have created the potential to produce patient-specific, biomimetic MPS for detailed mechanistic studies. It is projected that success in the development and maturation of patient-derived MPS with known genotypes and fully matured adult phenotypes will lead to advanced applications in precision medicine. In this Review, we examine human biomimetic liver MPS that are designed to recapitulate the liver acinus structure and functions to enhance our knowledge of the mechanisms of disease progression and of the absorption, distribution, metabolism, excretion and toxicity of therapeutic candidates and drugs as well as to evaluate their mechanisms of action and their application in precision medicine and preclinical trials.