The Kirchhoff equation with periodic boundary conditions offers a rich and elegant model for capturing the transverse oscillations of a nonlinear elastic medium. Since its origin in 1876, it has intrigued mathematicians worldwide with its intricate structure and challenging behavior. In this presentation, we explore the fundamental properties of the equation, motivated by both its physical background and deep mathematical significance. Key theoretical aspects are addressed, including local well-posedness and the lifespan of solutions, that are particularly subtle due to the equation's quasi-linear nature. A central component of our approach is the Hamiltonian formulation of the problem, which provides a rigorous and insightful framework for analyzing the solution dynamics. Most notably, we place special emphasis on the role of normal form techniques, which serve as a cornerstone of our analysis. These methods are crucial in overcoming the difficulties posed by the nonlinearities and in gaining a deeper understanding of the long-time behavior of solutions. Our aim is to offer a clear and structured overview of the current challenges, with a particular focus on how normal form transformations can be used to address them effectively.