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Almost 50 years have passed from the first computer simulations of water, and a large number of molecular models have been proposed since then to elucidate the unique behavior of water across different phases. In this article, we review the recent progress in the development of analytical potential energy functions that aim at correctly representing many-body effects. Starting from the many-body expansion of the interaction energy, specific focus is on different classes of potential energy functions built upon a hierarchy of approximations and on their ability to accurately reproduce reference data obtained from state-of-the-art electronic structure calculations and experimental measurements. We show that most recent potential energy functions, which include explicit short-range representations of two-body and three-body effects along with a physically correct description of many-body effects at all distances, predict the properties of water from the gas to the condensed phase with unprecedented accuracy, thus opening the door to the long-sought "universal model" capable of describing the behavior of water under different conditions and in different environments., Funding Agencies|Royal Swedish Academy of Sciences through Nobel Institutes for Physics and Chemistry; Swedish Research Council; Department of Physics at Stockholm University; Icelandic Research Fund; Army Research Laboratory [W911NF-12-2-0023]; Cluster of Excellence RESOLV - Deutsche Forschungsgemeinschaft (DFG) [EXC 1069]; Leverhulme Early Career Fellowship [1441]; Isaac Newton Trust; Wayne State University; National Institutes of Health [R01GM108583]; National Science Foundation [CHE-1453204, ACI-1053575]; National Energy Research Scientific Computing Center (NERSCC); Office of Science of the U.S. Department of Energy [DE-AC02-05CH11231]