Hsp70 molecular chaperones play central roles in maintaining a healthy cellular proteome. Hsp70s function by binding to short peptide sequences in incompletely folded client proteins, thus preventing them from misfolding and/or aggregating, and in many cases holding them in a state that is competent for subsequent processes like translocation across membranes. There is considerable interest in predicting the sites where Hsp70s may bind their clients, as the ability to do so sheds light on the cellular functions of the chaperone. In addition, the capacity of the Hsp70 chaperone family to bind to a broad array of clients and to identify accessible sequences that enable discrimination of those that are folded from those that are not fully folded, which is essential to their cellular roles, is a fascinating puzzle in molecular recognition. In this article we discuss efforts to harness computational modeling with input from experimental data to develop a predictive understanding of the promiscuous yet selective binding of Hsp70 molecular chaperones to accessible sequences within their client proteins. We trace how an increasing understanding of the complexities of Hsp70-client interactions has led computational modeling to new underlying assumptions and design features. We describe the trend from purely data-driven analysis toward increased reliance on physics-based modeling that deeply integrates structural information and sequence-based functional data with physics-based binding energies. Notably, new experimental insights are adding to our understanding of the molecular origins of "selective promiscuity" in substrate binding by Hsp70 chaperones and challenging the underlying assumptions and design used in earlier predictive models. Taking the new experimental findings together with exciting progress in computational modeling of protein structures leads us to foresee a bright future for a predictive understanding of selective-yet-promiscuous binding exploited by Hsp70 molecular chaperones; the resulting new insights will also apply to substrate binding by other chaperones and by signaling proteins.