  Magnetism is one of the largest, most fundamental, and technologically most
relevant fields of condensed-matter physics. Traditionally, two basic magnetic
phases have been considered -- ferromagnetism and antiferromagnetism. The
breaking of the time-reversal symmetry and spin splitting of the electronic
states by the magnetization in ferromagnets underpins a range of macroscopic
responses in this extensively explored and exploited type of magnets. By
comparison, antiferromagnets have vanishing net magnetization. This Perspective
reflects on recent observations of materials hosting an intriguing
ferromagnetic-antiferromagnetic dichotomy, in which spin-split spectra and
macroscopic observables, akin to ferromagnets, are accompanied by antiparallel
magnetic order with vanishing magnetization, typical of antiferromagnets. An
unconventional non-relativistic symmetry-group formalism offers a resolution of
this apparent contradiction by delimiting a third basic magnetic phase, dubbed
altermagnetism. Our Perspective starts with an overview of the still emerging
unique phenomenology of the phase, and of the wide array of altermagnetic
material candidates. In the main part of the article, we illustrate how
altermagnetism can enrich our understanding of overarching condensed-matter
physics concepts, and have impact on prominent condensed-matter research areas.
