Abstract
A recent discovery has pointed out that the moment of inertia, the tidal Love number and the spin- induced quadrupole moment of a neutron star, are linked together by semi-analytic relations which are almost insensitive to the star equations of state. These I-Love-Q relations open the opportunity to combine gravitational and electromagnetic observations and to devise new strategies, which will provide a better insight on the neutron star structure and the nature of gravity in the strong field regime. In this paper we review the main features of these universal relations, studying four scenarios in which they have been analysed to extend their domain of validity. We focus first on compact binaries close to the coalescence. Then, we investigate how strong magnetic fields, fast rotation, and hot equations of state affect the I-Love-Q universality
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