  A number $\lambda \in \mathbb C $ is called an {\it eigenvalue} of the matrix
polynomial $P(z)$ if there exists a nonzero vector $x \in \mathbb C^n$ such
that $P(\lambda)x = 0$. Note that each finite eigenvalue of $P(z)$ is a zero of
the characteristic polynomial $\det(P(z))$. In this paper we establish some
(upper and lower) bounds for eigenvalues of matrix polynomials based on the
norm of their coefficient matrices and compare these bounds to those given by
N.J. Higham and F. Tisseur, J. Maroulas and P. Psarrakos.
