We consider the fluctuation of linear eigenvalue statistics of random band $n$ dimensional matrices whose bandwidth $b$ is assumed to grow with n in such a way that $b/n$ tends to zero. 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 remove the main technical restriction $n>>b>>n^{1/2}$ of all the papers, in which band matrices were studied before. Moreover, the developed method 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.