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Our general and operative notion of chemistry, that enabling us to describe and interpret the structural, bonding and reactivity properties of the various compounds, is essentially rooted in the periodicity of properties of their constituting elements. Periodicity drove the path towards the proposition of the periodic table of elements and its rationalization thanks to the gradually acquired knowledge on the atomic electronic structure. Atomic radii periodicity mirrors the periodicity of the electronic configurations of the atoms’ outermost shell. The changes in atomic radii differentiate the chemical properties, hence the structures and properties of the elemental solids and of their compounds. Yet, under an external pressure, such a fully rationalized scenario may drastically change. When an atom is compressed, its average electron density increases and its outermost electronic shell is the easiest one to compress. At 100 Giga Pascal (GPa), that is at 106 atmospheres, the change of the atomic radius along a period of the Periodic Table becomes much less evident and at 1000 GPa our classical notion of periodicity is completely lost. Under pressure, the energy due to the compression work made on a system adds up to its internal energy. With such an energy gain the system may reach regions of the potential energy surface which would not be otherwise accessible. The chemical bond nature may change, even radically, and new structures and bonding patterns, characterized by totally unexpected properties, become energetically stable and possible. For instance, sodium, which is a silvery-white, highly reactive metal, becomes a fully transparent insulator, while boron is turned into a partially ionic solid elemental phase because charge transfer takes place between differently clustered groups of boron atoms. The incredible chemical inertness of helium finally falls as it forms a stable compound with sodium, Na2He. Under a suitable pressure, compounds with unusual stoichiometry (Na3Cl2, Na2Cl, Na3Cl, NaCl3, NaCl7) may be observed, despite their formula would be immediately rejected if proposed by a student at any high-school or university exam, or new carbon allotropes may appear or the aromatic character of benzene may vanish. This new chemistry is usually predicted through ab-initio quantum mechanical methods and interpreted and rationalized with the most modern chemical bonding approaches. However, compounds anticipated in silico have then be reproduced experimentally in many cases, by using diamond anvil cells to synthesize them and a variety of in situ instrumental techniques to characterize them properly.