A hybrid energy-storage system (HESS), which fully utilizes the durability of energy-oriented storage devices and the rapidity of power-oriented storage devices, is an efficient solution to managing energy and power legitimately and symmetrically. Hence, research into these systems is drawing more attention with substantial findings. A battery–supercapacitor hybrid energy-storage system (BS-HESS) is widely adopted in the fields of renewable energy integration, smart- and micro-grids, energy integration systems, etc. Focusing on the BS-HESS, in this work we present a comprehensive survey including technologies of the battery management system (BMS), power conversion system (PCS), energy management system (EMS), predictive control techniques of the underlying system, application and cost-effective feasibility aspects, etc. This work reflects strong symmetry on different aspects of designing an HESS, and provides guidelines and design references for the research and application of an HESS.