  Moir\'e superlattices in van der Waals structures can be used to control the
electronic properties of the material and lead to emergent correlated and
topological phenomena. Its first demonstration in van der Waals magnets
exhibited noncollinear states and domain structures with, however, limited
manipulation. Here we report electrically tunable moir\'e magnetism in twisted
double bilayers - that is, a bilayer plus a bilayer with a twist angle between
them - of layered antiferromagnet CrI3. Using magneto-optical Kerr effect
microscopy, we observe the coexistence of antiferromagnetic and ferromagnetic
order with nonzero net magnetization - a hallmark of moir\'e magnetism. Such
magnetic state extends over a wide range of twist angles (with transitions at
around 0{\deg} and above 20{\deg}) and exhibits a nonmonotonic temperature
dependence. We further demonstrate voltage-assisted magnetic switching. The
observed nontrivial magnetic states and unprecedented control by twist angle,
temperature and electrical gating are supported by the simulated phase diagram
of the moir\'e magnetism.
