  We propose to implement tunable higher-order topological states in a
heterojunction consisting of a two-dimensional (2D) topological insulator and
the recently discovered altermagnets, whose unique spin-polarization in both
real and reciprocal space and null magnetization are in contrast to
conventional ferromagnets and antiferromagnets. Based on symmetry analysis and
effective edge theory, we show that the special spin splitting in altermagnets
with different symmetries, such as $d$-wave, can introduce Dirac mass terms
with opposite signs on the adjacent boundaries of the topological insulator,
resulting in the higher-order topological state with mass-domain bound corner
states. Moreover, by adjusting the direction of the N\'{e}el vector, we can
manipulate such topological corner states by moving their positions. By
first-principles calculations, taking a 2D topological insulator bismuthene
with a square lattice on an altermagnet MnF$_2$ as an example, we demonstrate
the feasibility of creating and manipulating the higher-order topological
states through altermagnets. Finally, we discuss the experimental
implementation and detection of the tunable topological corner states, as well
as the potential non-Abelian braiding of the Dirac corner fermions.
