Abstract: The analytical continuation algorithm plays a vital role in constructing the global gravity field model, refining the local geoid, etc. The one-order continuation algorithm is widely adopted in conventional analytical continuation calculations, while the high-order counterpart is rarely applied. To enhance the accuracy of the analytical continuation algorithm, an in-depth investigation on the high-order analytical continuation algorithm is carried out in this paper. The calculation of the high-order radial derivatives is critical to the realization of the high-order analytical continuation algorithm. To begin with, the general formulae for the radial derivatives of harmonic function and gravity disturbance are deduced by the direct derivation method. Subsequently, the specific expressions for the radial derivatives of the first six orders based on direct derivation method are given according to the general formulae. Next, the radial derivatives of the first through sixth orders of harmonic function and gravity disturbance are deduced using the recurrence method. Obvious discrepancies exist between the formulae for the high-order radial derivatives obtained using the two derivation methods. Numerical tests demonstrate that the formulae for the high-order radial derivatives obtained by the direct derivation method have inherent theoretical defects, while those deduced via the recurrence method are theoretically sound and practically applicable. Therefore, the high-order analytical continuation algorithm should adopt the radial derivative formulae from the recurrence method rather than those from the direct derivation method. For quantitative assessment of the performance of the high-order analytical continuation algorithm, numerical experiments are carried out on full-degree model gravity data with a resolution of 2′×2′ using the formulae for the radial derivatives obtained via the recurrence method. The experimental results indicate that, under the noise perturbation of 1mGal, the measurement-surface continuation accuracies of the two-, three-, four-, five- and six-order analytical continuation algorithms are improved by 36.26%, 47.30%, 50.00%, 50.68% and 50.68% compared with the conventional one-order algorithm, respectively. Lastly, analytical continuation tests on the terrestrial observed gravity data are performed using the radial derivative formulae obtained using the recurrence method. The results show that the two-, three- and four-order continuation algorithms outperform the conventional one-order algorithm by 31.65%, 39.66% and 41.77% in terms of the continuation accuracy, respectively. Compared with the four-order analytical continuation algorithm, both the five- and six-order algorithms achieve only a marginal accuracy improvement of 0.01mGal. According to the findings of this study, the three- or four-order continuation is recommended for practical applications of analytical continuation algorithm.