We present some recent results obtained in collaboration with S.P. Novikov (Steklov Institute and University of Maryland). We study the spectral theory for ordinary differential operators with special singularities such that all eigenfunctions are locally meromorphic near all real singular points. Such operators are called spectrally-meromorphic. In particular, all singular finite-gap operators satisfy this condition. We show that for periodic spectrally-meromorphic operators the Bloch variety is well-defined, and this observation provides a natural way to show that at least locally our operators can be approximated by the finite-gap ones.