In conventional Computer Tomography (CT), reconstruction of even a small region of interest inside an object requires irradiating the entire object with x-rays. Recently a new group of algorithms has emerged that reconstructs an image from interior data, i.e. the data that is based only on x-rays intersecting the region of interest. In medical applications of CT, collecting interior data results in a reduction of the x-ray dose to the patient. In this talk we discuss some mathematical aspects of image reconstruction from interior data. Our starting point is the Gelfand-Graev formula, which establishes a relation between the ray transform of a function and its Hilbert transform along lines. Image reconstruction is reduced to the problem of inverting the finite Hilbert transform (FHT) with different types of incomplete data. In this talk we discuss several such problems and the progress made in solving them. Two key tools in our analysis are a differential operator that commutes with the FHT and the theory of the matrix Riemann-Hilbert problem.” “This is joint work with A. Tovbis and M. Bertola.