• stratified groups
• partial differentiability
• intrinsic differentiability
• homogeneous groups
• h-differentiability
• distributional derivatives
• Taylor inequality
• Taylor polynomials

The purpose of this thesis is to provide a concise and rigorous overview of the algebraic and metric structures of homogeneous and stratified groups, in order to develop some aspects of the theory of intrinsic differentiability on these spaces. We will define three notions of differentiability and we will give some characterizations, both in homogeneous groups and stratified groups.
We will then define the class of intrinsic regular functions and their intrinsic Taylor polynomials, showing, as an application, the corresponding Taylor formula.