We study qualitative and quantitative properties of the Laplacian and related operators, with an emphasis on low eigenvalues and properties of eigenfunctions. One major driving force of this research has been the relation between eigenvalues and geometric aspects such as explicit characterizations of the asymptotics of eigenvalues of thin domains, the effect of bending on eigenvalues, the location of nodal lines, etc. Recently, we have also began to explore the possibilities provided by some numerical methods to obtain insight into certain aspects of the theory. This has allowed us to formulate several new conjectures backed up by extensive simulation. The problems considered have several applications in Physics and Mechanics.