In this paper, the stability of matrix polynomials is investigated. First, upper and lower bounds are derived for the eigenvalues of a matrix polynomial. The bounds are based on the spectral radius and the norms of the related matrices, respectively. Then, by means of the argument principle, stability criteria are presented which are necessary and sufficient conditions for the stability of matrix polynomials. Furthermore, a numerical algorithm is provided for checking the stability of matrix polynomials. Numerical examples are given to illustrate the main results.