Eigenvalue and spectrum have many insightful applications in mathematics and applied sciences. Though a large theory has been developed, this topic has faced many challenging problems. For example, some results in the inverse spectral problems and our recent works have revealed out some very strong continuous dependence of Sturm-Liouville operators on potentials and weights. More precisely, eigenvalues are continuously dependent on potentials and weights even when the weak topology is considered. With this, we have recently solved several basic extremal problems on eigenvalues of Sturm-Liouville operators when the physical measurement for potentials are given. In doing so, we systematically developed some analytical methods toward the complete solution of these problems. The relevant results have been published on "J. Differential Equations 247 (2009) 364 - 400".