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We review the concept of purely virtual particle and its uses in quantum gravity, primordial cosmology and collider physics. The fake particle, or "fakeon", which mediates interactions without appearing among the incoming and outgoing states, can be introduced by means of a new diagrammatics. The renormalization coincides with the one of the parent Euclidean diagrammatics, while unitarity follows from spectral optical identities, which can be derived by means of algebraic operations. The classical limit of a theory of physical particles and fakeons is described by an ordinary Lagrangian plus Hermitian, micro acausal and micro nonlocal self-interactions. Quantum gravity propagates the graviton, a massive scalar field (the inflaton) and a massive spin-2 fakeon, and leads to a constrained primordial cosmology, which predicts the tensor-to-scalar ratio $r$ in the window $0.4\lesssim 1000r\lesssim 3.5$. The interpretation of inflation as a cosmic RG flow allows us to calculate the perturbation spectra to high orders in the presence of the Weyl squared term. In models of new physics beyond the standard model, fakeons evade various phenomenological bounds, because they are less constrained than normal particles. The resummation of self-energies reveals that it is impossible to get too close to the fakeon peak. The related peak uncertainty, equal to the fakeon width divided by 2, is expected to be observable.