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Hybrid-kinetic simulations describe ion-scale kinetic phenomena in space plasmas by considering ions kinetically, i.e. as particles, while electrons are modelled as a fluid. Most of the existing hybrid-kinetic codes neglect the electron mass (see chapter 3) for a simplified calculation of the electromagnetic fields. There are, however, situations in which delay in the electrons response due to the electron inertia matters. This chapter concentrates on hybrid-kinetic simulation models which take the finite mass of the electron fluid into account. First a review is given of the history of including the finite electron mass in hybrid-kinetic models. Then the equations are discussed which additionally have to be solved compared to the mass-less hybrid-kinetic models. For definiteness their numerical implementation without additional approximations is illustrated by describing a hybrid-kinetic code, CHIEF. The importance of the consideration of the finite electron mass are discussed for typical applications (magnetic reconnection, plasma turbulence, collisionless shocks and global magnetospheric simulations). In particular the problem of guide field magnetic reconnection is addressed in some detail. Possible next steps towards further improvements of hybrid-kinetic simulations with finite electron mass are suggested.