Abstract The integration of different energy systems is receiving increasing attention as a promising solution to accommodate the rising share of renewable energy sources (RES). The synergies among different energy carriers can be exploited to introduce flexibility to the system and compensate for the uncertain and fluctuating production from RES. In particular, the power and natural gas networks are becoming increasingly coupled due to the growth in electricity produced from gas-fired power plants, and the emergence of power-to-gas technologies. This paper aims at analyzing the existing literature about the short-term optimal operation of integrated electrical-gas systems (IEGSs) and identifying the benefits of coordinated optimization compared to independent scheduling of the two sectors. A comprehensive overview of the mathematical modeling of the two systems and the linking components is presented. The results show that lower operating costs and higher utilization of renewables are achieved by adopting fully integrated optimization strategies. Moreover, linepack modeling is crucial to fully capture the intrinsic storage capability of the gas network. Finally, modeling the uncertainties from RES production is imperative, since these are reflected in the gas network. Whilst significant research about IEGS modeling has been undertaken, there are still several challenges in solving the co-optimization problem. In this review, possible solution approaches are discussed, identifying linearization and convex relaxation techniques as powerful tools to approximate the nonconvex and nonlinear gas flow equation. Furthermore, decomposition techniques and decentralized optimization schemes can be used to more efficiently solve the problem, and, concurrently, tackle regulatory and privacy issues.