This paper proposes a 6D movable antenna (6DMA) system for future wireless networks to improve communication performance. Unlike traditional fixed-position antennas (FPAs) and existing 2D movable antennas (2DMA), the 6DMA system allows independent adjustment of both 3D positions and 3D rotations of distributed antenna surfaces within a given space. The 6DMA is applied to the base station (BS) to provide full degrees of freedom (DoFs) for adapting to dynamic user spatial distributions. The challenge is to optimally control the 6D positions and rotations of all 6DMA surfaces to maximize network capacity, subject to practical constraints on antenna movement. The paper models the 6DMA-enabled BS and user channels in terms of 6D positions and rotations of all 6DMA surfaces. An efficient alternating optimization algorithm is proposed to search for the best 6D positions and rotations by leveraging Monte Carlo simulation. The algorithm sequentially optimizes the 3D position/3D rotation of each 6DMA surface with those of the other surfaces fixed in an iterative manner. Numerical results show that the proposed 6DMA-BS significantly improves network capacity compared to benchmark BS architectures with FPAs or partially movable antennas, especially when user distribution is more spatially non-uniform. The 6DMA system offers greater flexibility in antenna deployment, allowing the BS to adapt to varying user distributions more effectively. The system includes practical constraints such as minimum distance between antenna surfaces, rotation constraints to avoid signal reflection and blockage, and the use of a Fibonacci Sphere-based random initialization scheme to ensure good performance of the converged solution. The paper also presents a detailed system model, channel model, and user distribution model, along with an optimization problem formulation and a proposed algorithm for solving it. The algorithm uses alternating optimization to maximize the approximate network capacity by jointly optimizing the 3D positions and 3D rotations of all 6DMA surfaces.This paper proposes a 6D movable antenna (6DMA) system for future wireless networks to improve communication performance. Unlike traditional fixed-position antennas (FPAs) and existing 2D movable antennas (2DMA), the 6DMA system allows independent adjustment of both 3D positions and 3D rotations of distributed antenna surfaces within a given space. The 6DMA is applied to the base station (BS) to provide full degrees of freedom (DoFs) for adapting to dynamic user spatial distributions. The challenge is to optimally control the 6D positions and rotations of all 6DMA surfaces to maximize network capacity, subject to practical constraints on antenna movement. The paper models the 6DMA-enabled BS and user channels in terms of 6D positions and rotations of all 6DMA surfaces. An efficient alternating optimization algorithm is proposed to search for the best 6D positions and rotations by leveraging Monte Carlo simulation. The algorithm sequentially optimizes the 3D position/3D rotation of each 6DMA surface with those of the other surfaces fixed in an iterative manner. Numerical results show that the proposed 6DMA-BS significantly improves network capacity compared to benchmark BS architectures with FPAs or partially movable antennas, especially when user distribution is more spatially non-uniform. The 6DMA system offers greater flexibility in antenna deployment, allowing the BS to adapt to varying user distributions more effectively. The system includes practical constraints such as minimum distance between antenna surfaces, rotation constraints to avoid signal reflection and blockage, and the use of a Fibonacci Sphere-based random initialization scheme to ensure good performance of the converged solution. The paper also presents a detailed system model, channel model, and user distribution model, along with an optimization problem formulation and a proposed algorithm for solving it. The algorithm uses alternating optimization to maximize the approximate network capacity by jointly optimizing the 3D positions and 3D rotations of all 6DMA surfaces.