  We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with
upper triangular entries being independent identically distributed random
variables with mean zero and unit variance. We additionally suppose that
$\mathbb E |X_{11}|^{4 + \delta} =: \mu_{4+\delta} < C$ for some $\delta > 0$
and some absolute constant $C$. Under these conditions we show that the typical
Kolmogorov distance between the empirical spectral distribution function of
eigenvalues of $n^{-1/2} {\bf X}$ and Wigner's semicircle law is of order $1/n$
up to some logarithmic correction factor. As a direct consequence of this
result we establish that the semicircle law holds on a short scale.
Furthermore, we show for this finite moment ensemble rigidity of eigenvalues
and delocalization properties of the eigenvectors. Some numerical experiments
are included illustrating the influence of the tail behavior of the matrix
entries when only a small number of moments exist.
