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Rare-earth-based barium copper oxide (REBCO) coated conductors are promising candidates for the development of ultra-high-field (UHF) magnets, due to its high in-field performance, engineering current density, tensile strength and commercial availability. However, technological challenges pertaining to the large screening currents still remain. The major issues caused by the screening currents in REBCO conductors in UHF applications involve two aspects: the screening current induced magnetic field (SCF), and the screening current induced stress (SCS). In the past decades, extensive research has been devoted to the SCF, offering a variety of possible remedies. With latest advances in the construction and testing of UHF magnets, new observations of the SCF involving REBCO coils were reported. The SCS was identified in recent years and has raised growing concerns. The excessive and highly concentrated Lorentz force, rooted in the high magnetic field and the screening currents, poses threats to the mechanical strength of REBCO coated conductors. The aim of this paper is to review recent research efforts in understanding and tackling the screening current related technological issues. For the SCF, we focus on the latest observations in high-field experiments and its various mitigation methods. For the SCS, we present recent studies including experimental characterizations, numerical modelling and possible countermeasures. It is still an open question to precisely predict the SCS in large-scale HTS magnets. How to minimize the influence of SCF and SCS is one of the key technical challenges for the design of future UHF magnets.