The paper is a beautiful and complete survey on old and recent results and open questions about minimization problems involving the eigenvalues of the Laplace operator with respect to variations of the domain. These problems can be written as \[ \min\{\Phi(\lambda_1(\Omega),\dots, \lambda_k(\Omega):\Omega\in{\mathcal A}\}, \] where \({\mathcal A}\) is the class of admissible domains and \(\lambda_j(\Omega)\) are the eigenvalues of the Dirichlet Laplacian on \(\Omega\). When the admissible domains are taken in a given bounded set, the existence of an optimal \(\Omega_{\text{opt}}\) follows from general results; on the contrary, the analysis is more delicate when the admissible \(\Omega\) may vary in the entire space \(\mathbb{R}^N\). Several results, conjectures and open problems are presented; in particular:
\(\bullet\) classical isoperimetric inequalities for the first two eigenvalues;
\(\bullet\) existence of optimal domains minimizing the third eigenvalue, and conjectures about its shape;
\(\bullet\) conjectures about higher eigenvalues;
\(\bullet\) results and open questions about minimization problems when the admissible domains are supposed convex;
\(\bullet\) results, conjectures and open questions about optimization problems for the Laplace operator with different boundary conditions (Neumann, Robin, Stekloff).