  Kaniadakis ($\kappa$-deformed) statistics is being widely used for describing
relativistic systems with non-extensive behavior and/or interactions. It is
built upon a one-parameter generalization of the classical
Boltzmann-Gibbs-Shannon entropy, possessing the latter as a particular
sub-case. Recently, Kaniadakis model has been adapted to accommodate the
complexities of systems under the influence of gravity. The ensuing framework
exhibits a rich phenomenology that allows for a deeper understanding of the
most extreme conditions of the Universe. Here we present the state-of-the-art
of $\kappa$-statistics, discussing its virtues and drawbacks at different
energy scales. Special focus is dedicated to gravitational and cosmological
implications, including effects on the expanding Universe in dark energy
scenarios. This review highlights the versatility of Kaniadakis paradigm,
demonstrating its broad application across various fields and setting the stage
for further advancements in statistical modeling and theoretical physics.
