Six-dimensional movable antenna (6DMA) is a new and revolutionary technique that fully exploits the wireless channel spatial variations at the transmitter/receiver by flexibly adjusting the three-dimensional (3D) positions and/or 3D rotations of antennas/antenna surfaces (sub-arrays), thereby improving the performance of wireless networks cost-effectively without the need to deploy additional antennas. It is thus expected that the integration of new 6DMAs into future sixth-generation (6G) wireless networks will fundamentally enhance antenna agility and adaptability, and introduce new degrees of freedom (DoFs) for system design. Despite its great potential, 6DMA faces new challenges to be efficiently implemented in wireless networks, including corresponding architectures, antenna position and rotation optimization, channel estimation, and system design from both communication and sensing perspectives. In this paper, we provide a tutorial on 6DMA-enhanced wireless networks to address the above issues by unveiling associated new channel models, hardware implementations and practical position/rotation constraints, as well as various appealing applications in wireless networks. Moreover, we discuss two special cases of 6DMA, namely, rotatable 6DMA with fixed antenna position and positionable 6DMA with fixed antenna rotation, and highlight their respective design challenges and applications. We further present prototypes developed for 6DMA-enhanced communication along with experimental results obtained with these prototypes. Finally, we outline promising directions for further investigation.