The general theme of the work presented in this book is at the crossroads of two fundamental areas of mathematics: the differential geometry and the Lie (super) algebras. In the recent years the interest concerns mostly the study of Lie (super) algebras and is confirmed by many works in this subject. The main applications are to mathematical physics and geometry. The overall objective of our research is to study the action of the Lie (super) algebra of vector fields on the (super) geometric quantities. This action is naturally and is linked to the equivariance notation compared to the change of coordinates.