When light waves propagate through the atmosphere, they experience varying degrees of wavefront aberration due to turbulence. Adaptive optical systems restore the distorted wavefront to a state close to the ideal light through the principle of phase conjugation, thereby achieving wavefront aberration correction. The low-order modes of Zernike polynomials expanded in a circular domain align with Seidel aberrations, and by generating conjugate phases through specific algorithms, effective wavefront aberration correction can be achieved. This paper systematically elaborates on the theory of Zernike polynomials and two types of adaptive optics systems-wavefront-sensing and wavefront-sensorless approaches. It provides a detailed analysis of the working principles of eight Zernike polynomial-based wavefront aberration correction algorithms and evaluates their correction performance, advantages, and limitations.