The saddlepoint method of asymptotic analysis provides accurate approximations to distributions of various test statistics, estimators and other probabilities arising in stochastic modeling of circular data. This chapter provides an original review of the saddlepoint approximation with circular data. This approximation is a large deviation approximation, thus substantially more accurate than the asymptotic Gaussian. A saddlepoint approximation is available for various nonparametric tests on the circle: goodness-of-fit and two-sample tests of equality of distributions. Saddlepoint approximations for computing P-values or power functions of optimal parametric tests of isotropy are also presented. Moreover, in planar random walks where the direction taken by a particle at each step follows a circular distribution, the distribution of the radial distance covered by the particle can be obtained by saddlepoint approximations.