  We study a large class of reversible Markov chains with discrete state space
and transition matrix $P_N$. We define the notion of a set of {\it metastable
points} as a subset of the state space $\G_N$ such that (i) this set is reached
from any point $x\in \G_N$ without return to x with probability at least $b_N$,
while (ii) for any two point x,y in the metastable set, the probability
$T^{-1}_{x,y}$ to reach y from x without return to x is smaller than
$a_N^{-1}\ll b_N$. Under some additional non-degeneracy assumption, we show
that in such a situation: \item{(i)} To each metastable point corresponds a
metastable state, whose mean exit time can be computed precisely. \item{(ii)}
To each metastable point corresponds one simple eigenvalue of $1-P_N$ which is
essentially equal to the inverse mean exit time from this state. The
corresponding eigenfunctions are close to the indicator function of the support
of the metastable state. Moreover, these results imply very sharp uniform
control of the deviation of the probability distribution of metastable exit
times from the exponential distribution.
