Traditionally, the primary field, where curvature has been at the heart of research, is the theory of general relativity. In recent studies, however, the impact of curvilinear geometry enters various disciplines, ranging from solid‐state physics over soft‐matter physics, chemistry, and biology to mathematics, giving rise to a plethora of emerging domains such as curvilinear nematics, curvilinear studies of cell biology, curvilinear semiconductors, superfluidity, optics, 2D van der Waals materials, plasmonics, magnetism, and superconductivity. Here, the state of the art is summarized and prospects for future research in curvilinear solid‐state systems exhibiting such fundamental cooperative phenomena as ferromagnetism, antiferromagnetism, and superconductivity are outlined. Highlighting the recent developments and current challenges in theory, fabrication, and characterization of curvilinear micro‐ and nanostructures, special attention is paid to perspective research directions entailing new physics and to their strong application potential. Overall, the perspective is aimed at crossing the boundaries between the magnetism and superconductivity communities and drawing attention to the conceptual aspects of how extension of structures into the third dimension and curvilinear geometry can modify existing and aid launching novel functionalities. In addition, the perspective should stimulate the development and dissemination of research and development oriented techniques to facilitate rapid transitions from laboratory demonstrations to industry‐ready prototypes and eventual products.

The concept of curvature and chirality in space and time are foundational for the understanding of the organic life and formation of matter in the Universe. Chiral interactions but also curvature effects are tacitly accepted to be local. A prototypical condensed matter example is a local spin-orbit- or curvature-induced Rashba or Dzyaloshinskii-Moriya interactions. Here, we introduce a chiral effect, which is essentially nonlocal and resembles itself even in static spin textures living in curvilinear magnetic nanoshells. Its physical origin is the nonlocal magnetostatic interaction. To identify this interaction, we put forth a self-consistent micromagnetic framework of curvilinear magnetism. Understanding of the nonlocal physics of curved magnetic shells requires a curvature-induced geometrical charge, which couples the magnetic sub-system with the curvilinear geometry. The chiral interaction brings about a nonlocal chiral symmetry breaking effect: it introduces handedness in an intrinsically achiral material and enables the design of magnetolectric and ferrotoroidic responses.

We show that the interaction of the magnetic subsystem of a curved magnet with the magnet curvature results in the coupling of a topologically nontrivial magnetization pattern and topology of the object. The mechanism of this coupling is explored and illustrated by an example of a ferromagnetic Möbius ring, where a topologically induced domain wall appears as a ground state in the case of strong easy-normal anisotropy. For the Möbius geometry, the curvilinear form of the exchange interaction produces an additional effective Dzyaloshinskii-like term which leads to the coupling of the magnetochirality of the domain wall and chirality of the Möbius ring. Two types of domain walls are found, transversal and longitudinal, which are oriented across and along the Möbius ring, respectively. In both cases, the effect of magnetochirality symmetry breaking is established. The dependence of the ground state of the Möbius ring on its geometrical parameters and on the value of the easy-normal anisotropy is explored numerically.

Antiferromagnetic insulators are a prospective materials platform for magnonics, spin superfluidity, THz spintronics, and non-volatile data storage. A magnetomechanical coupling in antiferromagnets offers vast advantages in the control and manipulation of the primary order parameter yet remains largely unexplored. Here, we discover a new member in the family of flexoeffects in thin films of Cr₂O₃. We demonstrate that a gradient of mechanical strain can impact the magnetic phase transition resulting in the distribution of the Néel temperature along the thickness of a 50-nm-thick film. The inhomogeneous reduction of the antiferromagnetic order parameter induces a flexomagnetic coefficient of about 15 μ_B nm⁻². The antiferromagnetic ordering in the inhomogeneously strained films can persist up to 100 °C, rendering Cr₂O₃ relevant for industrial electronics applications. Strain gradient in Cr₂O₃ thin films enables fundamental research on magnetomechanics and thermodynamics of antiferromagnetic solitons, spin waves and artificial spin ice systems in magnetic materials with continuously graded parameters.

Low‐dimensional magnetic architectures including wires and thin films are key enablers of prospective ultrafast and energy efficient memory, logic, and sensor devices relying on spin‐orbitronic and magnonic concepts. Curvilinear magnetism emerged as a novel approach in material science, which allows tailoring of the fundamental anisotropic and chiral responses relying on the geometrical curvature of magnetic architectures. Much attention is dedicated to magnetic wires of Möbius, helical, or DNA‐like double helical shapes, which act as prototypical objects for the exploration of the fundamentals of curvilinear magnetism. Although there is a bulk number of original publications covering fabrication, characterization, and theory of magnetic wires, there is no comprehensive review of the theoretical framework of how to describe these architectures. Here, theoretical activities on the topic of curvilinear magnetic wires and narrow nanoribbons are summarized, providing a systematic review of the emergent interactions and novel physical effects caused by the curvature. Prospective research directions of curvilinear spintronics and spin‐orbitronics are discussed, the fundamental framework for curvilinear magnonics are outlined, and mechanically flexible curvilinear architectures for soft robotics are introduced.