The “supergravity pdf” file embedded below presents the four-dimensional N = 1 supergravity theory and the four-dimensional N = 2 gauged supergravity theory. In the Hilbert-Einstein action for gravity, the gravitational field is present as the metric that defines the Levi-Civita connection which plays the role of a gauge field for gravity with respect to general coordinate transformation invariance of the action, having the Riemann tensor as field strenght. In supergravity theory fermions are involved (the spin 3/2 gravitino is the fermionic supersymmetric partner of the spin 2 graviton) and we have beside the general coordinate transform invariance also a local Lorentz invariance of special relativity so that gravity is present in the action as the vielbein having a “curved” index acted upon by general coordinate transformations and a newly introduced “flat” index acted upon by local Lorentz transformations. The gravitino field comes with the Rarita-Schwinger action and is the supersymmetric fermionic partner of the graviton field represented by the vielbein and at the same time the gauge field of the localized supersymmetry transformation of fields. While the Levi-Civita connection defines the derivation rule for the “curved” index, the newly introduced spin connection defines the derivation rule for the “flat” index and fermions that are Majorana spinors in the local Lorentz frame. We prove the invariance of the supergravity action under supersymmetry variations of the fields in the first order formalism where the spin connection is considered a independent field, as well as in the second order formalism where the spin connection depends on the vielbein and fermion fields by the Euler-Lagrange equations of motion. In the gauged supergravity theory we have a covariant vector gauge field for the invariance of the theory under local transformations between the two gravitino fields, matching the bosonic and fermionic on-shell degrees of freedom and we prove the supersymmetry invariance of the action under gauged supersymmetry variations, introducing naturally a cosmological constant term in the action. Finally we show that the cosmological constant term can arise only in a gauged supergravity theory, since the restrictions we have to impose on the supersymmetry transformation parameter are not suitable for Majorana spinors in a ungauged N = 1 theory.