The use of moiré patterns to manipulate two-dimensional materials has facilitated new possibilities for controlling material properties. The moiré patterns in the two-dimensional magnets can cause peculiar spin texture, as shown by previous studies focused on twisted bilayer systems. In our study, we develop a theoretical model to investigate the magnetic structure of twisted trilayer magnets. Unlike the twisted bilayer, the twisted trilayer magnet has four different local stacking structures distinguished by the interlayer couplings between the three layers. Our results show that the complex interlayer coupling effects in the moiré superlattice can lead to the stabilization of rich magnetic domain structures; these structures can be significantly manipulated by adjusting the twist angle. Additionally, external magnetic fields can easily manipulate these domain structures, indicating potential applications in spintronics devices.