The local regime of the random matrix theory deals with the behavior of eigenvalues of $n\times n$ random matrices on the intervals whose length is of the order of the mean distance between nearest eigenvalues. According to the Dyson universality conjecture the behavior does not depend on the matrix probability law (ensemble) and may only depend on the type of matrices (real symmetric, hermitian, or quaternion real in the case of real eigenvalues and orthogonal, unitary or symplectic in the case of the eigenvalues on the unit circle). This conjecture is one of the central questions in the random matrix theory and has a lot of physical application, but it still proved only for some special cases. In the talk we will discuss the universality conjecture in more details, as well as the methods allow to obtain its proof for certain classes of ensembles of random matrices.