Moiré superlattices of twisted nonmagnetic two-dimensional (2D) materials are highly controllable platforms for the engineering of exotic correlated and topological states. Here, we report emerging magnetic textures in small-angle twisted 2D magnet chromium triiodide (CrI3). Using single-spin quantum magnetometry, we directly visualized nanoscale magnetic domains and periodic patterns, a signature of moiré magnetism, and measured domain size and magnetization. In twisted bilayer CrI3, we observed the coexistence of antiferromagnetic (AFM) and ferromagnetic (FM) domains with disorder-like spatial patterns. In twisted double-trilayer CrI3, AFM and FM domains with periodic patterns appear, which is in good agreement with the calculated spatial magnetic structures that arise from the local stacking-dependent interlayer exchange interactions in CrI3 moiré superlattices. Our results highlight magnetic moiré superlattices as a platform for exploring nanomagnetism.