Heat diffusion, wave propagation (drums, guitars), electric and magnetic phenomena, quantum stuff, fluid flow... all these physical phenomena share the property of being modelled by partial differential equations (PDEs) where a major role is played by the Laplace operator, thus understanding its properties is very important. One main feature is that it can be "diagonalized", which means solving the PDE (called Hemholtz equation) −∆ψ = λψ, with boundary conditions that depend on the physical context, where both the number λ and the function ψ are unknowns. These two entities dictate much of the behavior of those physical problems. All these things depend very strongly on the shape of the domain where the problem is posed, so a number of questions arise: how does one compute λ, and what are the domains of a given volume that minimize/maximize it? How does λ depend on the size of the domain? Can we find general rules to estimate λ? Does λ have any general properties? What is the behavior of ψ in terms of the sign? How do different solutions (λ, ψ) compare to each other? What is the influence of the boundary conditions on all this stuff? What do all these words even mean? To answer these questions, a number of very interesting theories were invented, and very curious results were proved. This talk, besides introducing the main concepts, aims at exposing these surprising results and apparatus, avoiding technicalities (a.k.a. proofs) but discussing the ideas instead, as well as bringing you into contact some background instruments that should be part of the toolbox of any person interested in analysis. We will also explore some physical examples.