  This paper consists of four parts. In the first part, we explain what
eigenvalues we are interested in and show the difficulties of the study on the
first (non-trivial) eigenvalue through examples. In the second part, we present
some (dual) variational formulas and explicit bounds for the first eigenvalue
of Laplacian on Riemannian manifolds or Jacobi matrices (Markov chains). Here,
a probabilistic approach--the coupling methods is adopted. In the third part,
we introduce recent lower bounds of several basic inequalities; these are based
on a generalization of Cheeger's approach which comes from Riemannian geometry.
In the last part, a diagram of nine different types of ergodicity and a table
of explicit criteria for them are presented. These criteria are motivated by
the weighted Hardy inequality which comes from Harmonic analysis.
