In this review we discuss the latest results concerning development of the machine learning algorithms for characterization of the magnetic skyrmions that are topologically-protected magnetic textures originated from the Dzyaloshinskii-Moriya interaction that competes Heisenberg isotropic exchange in ferromagnets. We show that for classical spin systems there is a whole pool of machine approaches allowing their accurate phase classification and quantitative description on the basis of few magnetization snapshots. In turn, investigation of the quantum skyrmions is a less explored issue, since there are fundamental limitations on the simulation of such wave functions with classical supercomputers. One needs to find the ways to imitate quantum skyrmions on near-term quantum computers. In this respect, we discuss implementation of the method for estimating structural complexity of classical objects for characterization of the quantum skyrmion state on the basis of limited number of bitstrings obtained from the projective measurements.