  Block Toeplitz and Hankel matrices arise in many aspects of applications. In
this paper, we will research the distributions of eigenvalues for some models
and get the semicircle law. Firstly we will give trace formulae of block
Toeplitz and Hankel matrix. Then we will prove that the almost sure limit
$\gamma_{_T}^{(m)}$ $(\gamma_{_H}^{(m)})$ of eigenvalue distributions of random
block Toeplitz (Hankel) matrices exist and give the moments of the limit
distributions where $m$ is the order of the blocks. Then we will prove the
existence of almost sure limit of eigenvalue distributions of random block
Toeplitz and Hankel band matrices and give the moments of the limit
distributions. Finally we will prove that $\gamma_{_T}^{(m)}$
$(\gamma_{_H}^{(m)})$ converges weakly to the semicircle law as $m\ra\iy$.
