  Bringing magnetic metals into superconducting states represents an important
approach for realizing unconventional superconductors and potentially even
topological superconductors. Altermagnetism, classified as a third basic
collinear magnetic phase, gives rise to intriguing momentum-dependent
spin-splitting of the band structure, and results in an even number of
spin-polarized Fermi surfaces due to the symmetry-enforced zero net
magnetization. In this work, we investigate the effect of this new magnetic
order on the superconductivity of a two-dimensional metal with d-wave
altermagnetism and Rashba spin-orbital coupling. Specifically we consider an
extended attractive Hubbard interaction, and determine the types of
superconducting pairing that can occur in this system and ascertain whether
they possess topological properties. Through self-consistent mean-field
calculations, we find that the system in general favors a mixture of
spin-singlet s-wave and spin-triplet p-wave pairings, and that the
altermagnetism is beneficial to the latter. Using symmetry arguments supported
by detailed calculations, we show that a number of topological superconductors,
including both first-order and second-order ones, can emerge when the p-wave
pairing dominates. In particular, we find that the second-order topological
superconductor is enforced by a $\mathcal{C}_{4z}\mathcal{T}$ symmetry, which
renders the spin polarization of Majorana corner modes into a unique entangled
structure. Our study demonstrates that altermagnetic metals are fascinating
platforms for the exploration of intrinsic unconventional superconductivity and
topological superconductivity.
