We review generalized Fluctuation-Dissipation Relations which are valid under general conditions even in "nonstandard systems", e.g. out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperperturbed dynamics. In these relations, typically one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in non-standard cases, including driven granular media, systems with a multiscale structure, active matter and systems showing anomalous diffusion.The Fluctuation-Dissipation Theorem is a central result in equilibrium statistical mechanics. It allows one to express the linear response of a system to an external perturbation in terms of the spontaneous correlations, and its derivation is based on the detailed balance condition. In the last decades a great effort has been devoted to generalize this fundamental result to systems where detailed balance does not hold, because of the presence of some forms of dissipation, or energy and particle currents, that induce nonequilibrium conditions. Among the huge variety of systems belonging to this class, let us mention active and biological matter, driven granular media, molecular motors, and slow relaxing glasses. The derivation of a generalized Fluctuation-Dissipation Relation, and the investigation of its peculiar features in non-standard systems play therefore a central role in the building of a general theory of statistical mechanics beyond equilibrium.