This talk reports some recent study on the upper and lower bounds of eigenvalues by mixed finite element methods. Firstly, we build a theoretical framework, and show that certain finite element schemes can provide upper or lower bounds to the Stokes eigenvalues. Specifically, for these mixed element schemes, the loss of the local approximation property of the discrete velocity and pressure spaces may lead to different computed bounds of the eigenvalues. Formally theoretical analysis is constructed based on certain mathematical hypotheses, and numerical experiments are given to illustrate the validity of the theory. Then, by noting the asymptotic connection between the incompressible Stokes problem and the linear elasticity problem, we extend the discussions to linear elasticity eigenvalue problem, and lower and upper numerically computable bounds for the eigenvalues are derived based on certain mathematical hypotheses. Further, by utilizing the min-max principle and perturbation theory for the solution operator, theoretical lower and upper bounds can be controlled by setting proper Lam\'e parameters. If time permits, we will discuss some recent discoveries on the Laplace eigenvalue problems. These works are done jointly with Hongtao Chen and Yifan Yue.