Magnetically ordered crystals are traditionally divided into two basic phases -- ferromagnetism and antiferromagnetism. The ferromagnetic order offers a range of phenomena and device applications. The vanishing net magnetization in antiferromagnets is potentially favorable for spatial and temporal scalability of devices. Recently, our team and others have predicted instances of strong time-reversal symmetry breaking and spin splitting in electronic bands, typical of ferromagnetism, in crystals with antiparallel compensated magnetic order, typical of antiferromagnetism. Our central idea, resolving this apparent fundamental conflict in magnetism, is that symmetry classifies a third basic magnetic phase. Its alternating spin polarizations in both crystal-structure real space and electronic-structure momentum space suggest a term altermagnetism. We will demonstrate that altermagnets combine merits of ferromagnets and antiferromagnets, that were regarded as principally incompatible, and have merits unparalleled in either of the two traditional basic magnetic phases. In Objective 1 we will establish materials landscape of altermagnetism and, in Objective 2, show how its anisotropic (d-wave) nature enriches fundamental physics concepts of lifted Kramers spin-degeneracy, Fermi-liquid instabilities and electron quasiparticles. This will underpin our development of a new avenue in spintronics, elusive within the two traditional magnetic phases, based on strong non-relativistic spin-conserving phenomena, without magnetization imposed scalability limitations, and with complex functionalities. In Objective 3 we will demonstrate altermagnetic giant-magnetoresistive multi-layer memory devices. In Objective 4 we will realize logic-in-memory analog functionalities in single-layer nano-scale devices via atomically-sharp altermagnetic domain walls, written by pulses scaled down to femtoseconds, and forming networks of self-assembled atomic-scale giant-magnetoresistive junctions.