| Summary: | We present a new kind of model, which we call <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> Theories, where standard theories are modified including new fields, motivated by an additional symmetry (<inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> symmetry). In previous works, we proved that <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> Theories just live at one loop, so the model in a quantum level can be interesting. In the gravitational case, we have <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> Gravity, based on two symmetric tensors, <inline-formula> <math display="inline"> <semantics> <msub> <mi>g</mi> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mover accent="true"> <mi>g</mi> <mo stretchy="false">˜</mo> </mover> <mrow> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </semantics> </math> </inline-formula>, where quantum corrections can be controlled. In this paper, a review of the classical limit of <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> Gravity in a Cosmological level will be developed, where we explain the accelerated expansion of the universe without Dark Energy and the rotation velocity of galaxies by the Dark Matter effect. Additionally, we will introduce other phenomenon with <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> Gravity like the deflection of the light produced by the sun, the perihelion precession, Black Holes and the Cosmological Inflation.
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