On algebraic supergroups and quantum deformations

We give the definitions of affine algebraic supervariety and affine algebraic group through the functor of points and we relate them to the other definitions present in the literature. We study in detail the algebraic supergroup $SL(m|n)$ and give explicitly the Hopf algebra structure of the algebra representing its functor of points. At the end we give also the quantization of $SL(m|n)$ together with its coactions on suitable quantum spaces according to Manin's philosophy.