It is important to find upper and lower bounds of eigenvalues in the mathematical science and engineering applications. The minimum-maximum principle insures that numerical eigenvalues obtained by conforming finite elements approximate the exact ones from above. The computation of lower bounds is of interest. In this talk, I will present some of our work on asymptotic/guaranteed lower eigenvalue bounds, which is based on nonconforming finite element. For example, the lower eigenvalue bounds of elliptic eigenvalue problem with variable coefficients, Stokes eigenvalue problem and spectral problems arising in fluid mechanics, etc. Finally, I will discuss some recent ideas on lower bounds for eigenvalues.