Abstract: As an attempt to find a suitable replacement of the spectral measure of self-ajoint Jacobi operator when the self-adjointness is relaxed, we introduce a spectral data of a non-self-adjoint Jacobi operator. We show that the spectral map, which assigns to the Jacobi operator its spectral data, is bijective. This provides explicit answers to the direct and the inverse spectral problem for a class of non-self-adjoint Jacobi operators. The corresponding non-standard orthogonal polynomials will be introduced. If time permits, we also discuss a more conceptual framework of this theory from point of view of general anti-linear operators. The talk is based on a joint work with A. Pushnitski.