In this paper, we present a new wireless sensing system utilizing a movable antenna (MA) that continuously moves and receives sensing signals to enhance sensing performance over the conventional fixed-position antenna (FPA) sensing. We show that the angle estimation performance is fundamentally determined by the MA trajectory, and derive the Cramer-Rao bound (CRB) of the mean square error (MSE) for angle-of-arrival (AoA) estimation as a function of the trajectory for both one-dimensional (1D) and two-dimensional (2D) antenna movement. For the 1D case, a globally optimal trajectory that minimizes the CRB is derived in closed form. Notably, the resulting CRB decreases cubically with sensing time in the time-constrained regime, whereas it decreases linearly with sensing time and quadratically with the movement line segment's length in the space-constrained regime. For the 2D case, we aim to achieve the minimum of maximum (min-max) CRBs of estimation MSE for the two AoAs with respect to the horizontal and vertical axes. To this end, we design an efficient alternating optimization algorithm that iteratively updates the MA's horizontal or vertical coordinates with the other being fixed, yielding a locally optimal trajectory. Numerical results show that the proposed 1D/2D MA-based sensing schemes significantly reduce both the CRB and actual AoA estimation MSE compared to conventional FPA-based sensing with uniform linear/planar arrays (ULAs/UPAs) as well as various benchmark MA trajectories. Moreover, it is revealed that the steering vectors of our designed 1D/2D MA trajectories have low correlation in the angular domain, thereby effectively increasing the angular resolution for achieving higher AoA estimation accuracy.