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I will introduce the definition of non-locality based on the growth of the scattering amplitudes on the complex energy plane and discuss consistency of field theories with exponential-type amplitudes from the point of view of causality and unitarity. I will briefly review the possibility that gravity has unitary and non-local UV completion. It corresponds to infinite-derivative bare lagrangian which can make the theory renormalizable, at least at the level of power-counting. I will discuss the implications of this type of non-locality for the very possibility to have singular spherically symmetric black hole solutions. Another effect caused by the ghost-free condition and exponential form of the graviton propagator is related to the appearance of infinite number of complex-mass states on top of non-trivial background, such as de Sitter space or black hole. I will show that under certain conditions these states do not lead to instabilities at the level of perturbations, both in de Sitter space and around black hole.