The literature on eigenvalue problems is vast and so I am forced to be focused and biased in this survey talk. I will emphasize results that help to understand the strength of diffusion, the long term behavior of the corresponding evolution equation, and results that I perceive as directly useful for researchers who do nonlinear analysis such as stability and bifurcation analysis. Topics will include characterizations of eigenvalues, especially of the principal eigenvalue, of local and nonlocal operators, nodal property and multiplicity, effect of domain geometry on the size of the first nontrivial eigenvalue and shape optimization, effect of parameters such as diffusion and advection coefficients, nonlinear Krein-Rutman theory and its consequences to nonlinear systems. Through out the talk, I will mention open problems, some of which are well-known and long-standing, others new.