In the first part of the talk, I will introduce Wigner random matrices and describe a problem involving the spectrum of finite rank deformations of Wigner random matrices. In the second part, I will show how one can reduce this problem to a problem about the fluctuations of the matrix entries of regular functions of Wigner random matrices as the size of the matrix goes to infinity. The latter problem can be tackled by the CLT for martingale differences and some resolvent techniques. This is joint work with Alexander Soshnikov and David Renfrew.