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. CONTENTS I. Introduction 1 II. Altermagnetic phase 3 A. Ab initio band-structures 3 B. Symmetry classification and description 4 C. Identification rules 6 D. Material candidates 7 III. Physical concepts 9 A. Kramers theorem 9 B. Fermi-liquid instabilities 10 C. Electron and magnon quasiparticles 12 D. Berry phase and non-dissipative transport 14 IV. Research areas 15 A. Spintronics 15 B. Ultra-fast optics and neuromorphics 17 C. Thermoelectrics, field-effect electronics and multiferroics 19 D. Superconductivity 20 V. Conclusion 20 Acknowledgement 20 References 21