We propose an introduction to the differential geometry of connections on fiber bundles underlying the physics of (classical) gauge theories. Reminding first how Ehresmann connections are behind Yang-Mills-Utiyama theories, we will then be prepared to appreciate why Cartan connections provide a compelling geometric framework for gauge theories of gravity. A recipe of sort is then given, showing how these geometric data provide a kinematics that constrain and channel the construction of gauge theories, which specify the dynamics of the geometry. We end with a brief presentation of the “dressing field method”, a systematic tool to achieve gauge symmetry reduction that has a wide range of applications in gauge theory.