  We consider the fluctuation of linear eigenvalue statistics of random band
$n\times n$ matrices whose entries have the form
$\mathcal{M}_{ij}=b^{-1/2}u^{1/2}(|i-j|)\tilde w_{ij}$ with i.i.d. $w_{ij}$
possessing the $(4+\varepsilon)$th moment, where the function $u$ has a finite
support $[-C^*,C^*]$, so that $M$ has only $2C_*b+1$ nonzero diagonals. The
parameter $b$ (called the bandwidth) is assumed to grow with $n$ in a way that
$b/n\to 0$. Without any additional assumptions on the growth of $b$ we prove
CLT for linear eigenvalue statistics for a rather wide class of test functions.
Thus we improve and generalize the results of the previous papers [8] and [11],
where CLT was proven under the assumption $n>>b>>n^{1/2}$. Moreover, we develop
a method which allows to prove automatically the CLT for linear eigenvalue
statistics of the smooth test functions for almost all classical models of
random matrix theory: deformed Wigner and sample covariance matrices, sparse
matrices, diluted random matrices, matrices with heavy tales, etc.
