Abstract: The past decade has witnessed dramatic progress related to various aspects of emergent topological polar textures in oxide nanostructures displaying vortices, skyrmions, merons, hopfions, dipolar waves, or labyrinthine domains, among others. For a long time, these nontrivial structures (the electric counterparts of the exotic spin textures) were not expected due to the high energy cost associated with the dipolar anisotropy: the smooth and continuous evolution of the local polarization to produce topologically protected structures would result in a large elastic energy penalty. However, it was discovered that the delicate balance and intricate interplay between the electric, elastic, and gradient energies can be altered in low-dimensional forms of ferroelectric oxide nanostructures. These can be tuned to manipulate order parameters in ways once considered impossible. This review addresses the historical context that provided the fertile background for the dawning of the polar topological era. This has been possible thanks to a fruitful, positive feedback between theory and experiment: advances in materials synthesis and preparation (with a control at the atomic scale) and characterization have come together with great progress in theoretical modeling of ferroelectrics at larger length and timescales. An in-depth scientific description to formalize and generalize the prediction, observation, and probing of exotic, novel, and emergent states of matter is provided. Extensive discussions of the fundamental physics of such polar textures, a primer explaining the basic topological concepts, an explanation of the modern theoretical and computational methodologies that enable the design and study of such structures, what it takes to achieve deterministic, on-demand control of order-parameter topologies through atomically precise synthesis, the range of characterization methods that are key to probing these structures, and their thermodynamic field-driven (temperature-driven, stress-driven, etc.) susceptibilities are included. The new emergent states of matter join together with exotic functional properties (such as chirality, negative capacitance, and coexistence of phases) that, along with their small size and ultrafast dynamical response, make them potential candidates in multifunctional devices. Finally, some open questions and challenges for the future are presented, underlining the interesting future that is anticipated in this field.

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