Vortex beam is a kind of novel structured beam with helical wavefront and carries orbital angular momentum (OAM). Such structured field can find applications in many domains as large-capacity data transmission, remote detection, etc. The wavefront aberration occurs when the vortex beam propagates in a non-homogeneous medium as atmosphere turbulence, resulting in the OAM changing and go against practical applications. Therefore, it is necessary to compensate distorted vortex beams through adaptive optics. The recent advances on adaptive correction of distorted vortex beams was mainly reviewed. The current mature correction schemes were firstly introduced in brief, including wavefront sensing along with probe Gaussian beams, array detection along with phase retrieval algorithms, and so on. Then the deep-learning-based approaches were highlighted, as Zernike polynomial coefficients inversion, turbulence phase screen inversion, etc. The advantages and limitations of employing deep learning for distorted vortex beam compensation were also discussed. Finally, development trends of distortion compensation of vortex beams were prospected.