In the first part of the seminar, we will make a quick introduction to variational problems to motivate the interest in critical points. Then, we will focus on minimax critical points and introduce a useful tool to guarantee their existence: the Mountain Pass Theorem. We will start from the finite-dimensional setting, discussing a result by Courant, and arrive at the infinite-dimensional case, stating the classical Mountain Pass Theorem by Ambrosetti and Rabinowitz. We will try to stress the differences between the two cases and the need for compactness assumptions through counterexamples. In the second part of the seminar, we will show an application of the Mountain Pass Theorem to a semilinear elliptic problem with Neumann boundary conditions and, time permitting, we will try to face the lack of compactness in a particular case.