The properties of a magnetic state depend on which symmetries of the lattice leave the state unchanged when combined with time-reversal, i.e., with flipping all the magnetic moments. In a ferromagnet, no such symmetry exists, resulting in a nonzero magnetization and a uniform Zeeman splitting of the spin-up and spin-down bands. In contrast, this type of symmetry is present in a collinear antiferromagnet, since a lattice translation or inversion “undoes” the flipping of the spins, leading to degenerate spin-up and spin-down bands with no Zeeman splitting. Between these two types of magnetic states, however, lies a broad range of systems for which the symmetry that relates configurations of flipped spins is a rotation (proper or improper). Called altermagnets, these states have no magnetization, like an antiferromagnet, yet their bands display a nodal Zeeman splitting, resembling a "d-wave" (or higher-order) ferromagnet. In this talk, I will discuss the various connections between altermagnets and phenomena of interest in correlated electronic systems, such as multipolar order and Pomeranchuk instabilities. I will then show how spin-orbit coupling endows altermagnets with interesting and non-trivial topological properties, including mirror-protected nodal lines, Chern bands, and Weyl nodal lines in the electronic spectrum.