  We study the properties of operators in a unitary conformal field theory
whose scaling dimensions approach each other for some values of the parameters
and satisfy von Neumann-Wigner non-crossing rule. We argue that the scaling
dimensions of such operators and their OPE coefficients have a universal
scaling behavior in the vicinity of the crossing point. We demonstrate that the
obtained relations are in a good agreement with the known examples of the
level-crossing phenomenon in maximally supersymmetric $\mathcal N=4$ Yang-Mills
theory, three-dimensional conformal field theories and QCD.
