This chapter explores the Lagrangian formulation of electrodynamics and of relativistic matter fields, specifically focusing on complex scalar Minkowskian fields and their dynamics described through an action principle. The action is formulated as an integral over spacetime, where the Lagrangian density depends on the field and its derivatives, ensuring Lorentz invariance and yielding covariant field equations via the Euler-Lagrange equations. Crucial to maintaining Lorentz covariance is constructing terms, ensuring they form Lorentz scalars consistent with the field’s complex nature. The case of spinor fields is also considered, with special attention paid to the Foldy-Wouthuysen transformation. The chapter also goes into more depth into the use of differential forms, elucidating how these mathematical constructs encapsulate the geometric and algebraic structures essential for understanding the field equations derived from the action principle. Differential forms provide a powerful framework for expressing the underlying geometric properties and symmetries inherent in the field dynamics, enhancing the clarity and depth of the relativistic field theory discussed throughout the chapter.