  The enduring legacy of Euclidean geometry underpins classical machine
learning, which, for decades, has been primarily developed for data lying in
Euclidean space. Yet, modern machine learning increasingly encounters richly
structured data that is inherently nonEuclidean. This data can exhibit
intricate geometric, topological and algebraic structure: from the geometry of
the curvature of space-time, to topologically complex interactions between
neurons in the brain, to the algebraic transformations describing symmetries of
physical systems. Extracting knowledge from such non-Euclidean data
necessitates a broader mathematical perspective. Echoing the 19th-century
revolutions that gave rise to non-Euclidean geometry, an emerging line of
research is redefining modern machine learning with non-Euclidean structures.
Its goal: generalizing classical methods to unconventional data types with
geometry, topology, and algebra. In this review, we provide an accessible
gateway to this fast-growing field and propose a graphical taxonomy that
integrates recent advances into an intuitive unified framework. We subsequently
extract insights into current challenges and highlight exciting opportunities
for future development in this field.
