In the lectures delivered at the 2012 Predeal School an overview has been presented of the contemporary theory of the nuclear geometrical (shape) symmetries. The formalism combines two most powerful ...theory tools applicable in the context: The group- and group-representation theory together with the modern realistic mean-field theory. We suggest that all point-groups of symmetry of the mean-field Hamiltonian, sufficiently rich in symmetry elements (as discussed in the text) may lead to the magic numbers that characterise such a group in analogy with the spherical magic gaps characterising nuclear sphericity. We discuss in simple terms the mathematical and physical arguments for the presence of such symmetries in nuclei. In our opinion: It is not so much the question of Whether? – but rather: Where in the Nuclear Chart several of the point group-symmetries will be seen? We focus our presentation on the tetrahedral symmetry with the magic numbers calculated to be 32, 40, 56, 64, 70, 90 and 136, and discuss qualitatively the problem of the formulation of the experimental criteria which would allow for the final discovery of the tetrahedral symmetry in subatomic physics.
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