The present work reviews the results concerning quantum scattering theory of ultrarelativistic electrons in ultrathin crystals and its comparison with analogous classical results. It deals with an intermediate range of thicknesses, large enough for that the particle motion could not be considered as rectilinear but small enough for that the channeling regime of motion was not established. The quantum theory is based both upon the representation of the scattering amplitude as an integral over the surface surrounding the target, and on the so-called operator method of determination of the wave function as a solution of a Schrödinger-like equation. The latter method implies a wide use of the Fourier technique, both in calculation of each next step in the wave packet evolution, and in moving from the spatial coordinates to the angular ones. The authors compare the quantum differential scattering cross-sections with the classical ones in the considered range of crystal thicknesses and show their resemblances, distinctions and the evolution of these distinctions with the change of the particle energy. The simplest variant of quantum scattering theory based upon the eikonal approximation of quantum mechanics is considered. In the paper the quantum differential scattering cross-section was calculated and its affinity with the classical one was demonstrated. In the preparation of these lecture notes the material of the paper [4] was used.