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No-hair theorems do not allow for black holes in General Relativity (GR) to be characterized by any property other than their mass, electric charge, and spin. Scalar fields provide perhaps the simplest way to generalize GR, and it turns out that cases exist where the no-hair theorems can be evaded and black holes with scalar hair may emerge. In this talk I will examine the notion of nontrivial scalar field configurations arising in the strong regime of gravity near black holes, but also compact neutron stars and wormholes. I will go over the concept of spontaneous scalarization of compact objects and discuss the viability of scalarized solutions against observations.