In this paper, we propose an integrated sensing and communication (ISAC) system enabled by movable antennas (MAs), which can dynamically adjust antenna positions to enhance both sensing and communication performance for future wireless networks. To characterize the benefits of MA-enabled ISAC systems, we first derive the Cramér-Rao bound (CRB) for angle estimation error, which is then minimized for optimizing the antenna position vector (APV) and beamforming design, subject to a pre-defined signal-to-noise ratio (SNR) constraint to ensure the communication performance. In particular, for the case with receive MAs only, we provide a closed-form optimal antenna position solution, and show that employing MAs over conventional fixed-position antennas (FPAs) can achieve a sensing performance gain upper-bounded by 4.77 dB. On the other hand, for the case with transmit MAs only, we develop a boundary traversal breadth-first search (BT-BFS) algorithm to obtain the global optimal solution in the line-of-sight (LoS) channel scenario, along with a lower-complexity boundary traversal depth-first search (BT-DFS) algorithm to find a local optimal solution efficiently. While in the scenario with non-LoS (NLoS) channels, a majorization-minimization (MM) based Rosen's gradient projection (RGP) algorithm with an efficient initialization method is proposed to obtain stationary solutions for the considered problem, which can be extended to the general case with both transmit and receive MAs. Extensive numerical results are presented to verify the effectiveness of the proposed algorithms, and demonstrate the superiority of the considered MA-enabled ISAC system over conventional ISAC systems with FPAs in terms of sensing and communication performance trade-off.