  We propose magnetic second-order topological insulators (SOTIs). First, we
study a three-dimensional model. It is pointed out that the previously proposed
topological hinge insulator has actually surface states along the [001]
direction in addition to hinge states. We gap out these surface states by
introducing magnetization, obtaining a SOTI only with hinge states. The bulk
topological number is the $Z_2$ index protected by the combined symmetry of the
four-fold rotation and the inversion symmetry. We next study two dimensional
magnetic SOTIs, where the corner states are robust also in the presence of the
magnetization. Finally, we construct a magnetic second-order topological
semimetals by layering the two-dimensional magnetic SOTIs, where hinge-arc
states are robust also in the presence of the magnetization.
