The chapter delves deeper into advanced gauge field theory using fiber bundles and differential forms, beginning with the notation and symmetry of Christoffel symbols and their mixed indices. It summarizes gauge field theory by focusing on the covariant properties of matter fields under gauge transformations, introducing the covariant exterior derivative and its role in defining gauge covariant matter fields and the curvature 2-form. The exterior covariant derivative’s action on vector-valued and Lie-algebra-valued forms is detailed, followed by exercises to verify these concepts. The text then discusses the Yang-Mills theory in differential forms, presenting the Yang-Mills action, deriving the equations of motion, covering the Bianchi identity and gauge field equations, and briefly touching on the Abelian and non-Abelian cases. Overall, the chapter emphasizes the mathematical framework of fiber bundles and differential forms in gauge field theory, supplemented with exercises for deeper understanding.