One of the main goals of Riemannian geometry is to understand the relationship between the geometry and topology of smooth manifolds. To achieve this, it is useful to look for "canonical metrics" encoding the most significant topological information about a manifold. Among these metrics, we have the so-called critical metrics, which are critical points of specific curvature functionals. After introducing these objects, we will focus on the 4-dimensional case and explore the challenges associated with finding critical metrics through a direct variational approach. No previous knowledge will be assumed.