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We consider $n\times n$ real symmetric and Hermitian Wigner random matrices$n^{-1/2}W$ with independent (modulo symmetry condition) entries and the (null)sample covariance matrices $n^{-1}X^*X$ with independent entries of $m\times n$matrix $X$. Assuming first that the 4th cumulant (excess) $\kappa_4$ of entriesof $W$ and $X$ is zero and that their 4th moments satisfy a Lindeberg typecondition, we prove that linear statistics of eigenvalues of the above matricessatisfy the central limit theorem (CLT) as $n\to\infty$, $m\to\infty$, $m/n\toc\in[0,\infty)$ with the same variance as for Gaussian matrices if the testfunctions of statistics are smooth enough (essentially of the class$\mathbf{C}^5$). This is done by using a simple ``interpolation trick'' fromthe known results for the Gaussian matrices and the integration by parts,presented in the form of certain differentiation formulas. Then, by using amore elaborated version of the techniques, we prove the CLT in the case ofnonzero excess of entries again for essentially $\mathbb{C}^5$ test function.Here the variance of statistics contains an additional term proportional to$\kappa_4$. The proofs of all limit theorems follow essentially the samescheme.

Central limit theorem for linear eigenvalue statistics of random matrices with independent entries