  We systematically investigate the effect of the anomalous magnetic
moment(AMM) of quarks on the magnetized QCD matter, including the magnetic
susceptibility, the inverse magnetic catalysis around the critical temperature
and the neutral/charged pion and rho meson spectra under magnetic fields. The
dynamical AMM of quarks, its coupling with magnetic field causes Zeeman
splitting of the dispersion relation of quarks thus changes the magnetism
properties and meson mass spectra under magnetic fields. It is found that
including the AMM of quarks cannot fully understand lattice results of the
magnetized matter: The AMM of quarks reduces the dynamical quark mass thus
causes the inverse magnetic catalysis around $T_c$. The neutral pion mass is
very sensitive to the AMM, it decreases with magnetic field quickly, and the
charged pion mass shows a nonlinear behavior, i.e., firstly linearly increases
with the magnetic field and then saturates at strong magnetic field. For rho
meson, it is observed that AMM reduces the mass of neutral rho meson mass with
different $s_z$, and reduces the mass of $s_z=+1,0$ component charged rho meson
mass but enhances the $s_z=-1$ component charged rho meson mass. The magnetic
susceptibility at low temperature can be either positive or negative with
different AMM.
