Movable antenna (MA) technology exhibits great promise for enhancing the sensing capabilities of future sixth-generation (6G) networks due to its capability to alter antenna array geometry. With the growing prevalence of near-field propagation at ultra-high frequencies, this paper focuses on the application of one-dimensional (1D) and two-dimensional (2D) MA arrays for near-field sensing to jointly estimate the angle and distance information about a target. First, for the 1D MA array scenario, to gain insights into MA-enhanced near-field sensing, we investigate two simplified cases with only angle-of-arrival (AoA) or distance estimation, respectively, assuming that the other information is already known. The worst-case Cramer-Rao bounds (CRBs) on the mean square errors (MSEs) of the AoA estimation and the distance estimation are derived in these two cases. Then, we jointly optimize the positions of the MAs within the 1D array to minimize these CRBs and derive their closed-form solutions, which yield an identical array geometry to MA-enhanced far-field sensing. For the more challenging joint AoA and distance estimation, since the associated worst-case CRB is a highly complex and non-convex function with respect to the MA positions, a discrete sampling-based approach is proposed to sequentially update the MA positions and obtain an efficient suboptimal solution. Furthermore, we investigate the worst-case CRB minimization problems for a 2D MA array under various conditions and extend our proposed algorithms to solve them efficiently. Numerical results demonstrate that the proposed MA-enhanced near-field sensing scheme dramatically outperforms conventional fixed-position antennas (FPAs). Moreover, the joint angle and distance estimation results in a different array geometry from that in the individual estimation of angle/distance or far-field sensing.