  Polyhedral realization of crystal bases is one of the methods for describing
the crystal base $B(\infty)$ explicitly. This method can be applied to
symmetrizable Kac-Moody types. We can also apply this method to the crystal
bases $B(\lambda)$ of integrable highest weight modules and of modified quantum
algebras. But, the explicit forms of the polyhedral realizations of crystal
bases $B(\infty)$ and $B(\lambda)$ are only given in the case of arbitrary rank
2, of $A_n$ and of $A^{(1)}_n$. So, we will give the polyhedral realizations of
crystal bases $B(\infty)$ and $B(\lambda)$ for all simple Lie algebras in this
paper.
