Nonextensive Statistical Mechanics is a generalization of ordinary statistical mechanics,
based on the extremization of more general entropy measures than just the Shannon entropy.
These more general entropy measures are the so-called Tsallis entropies.
They depend on a real parameter q. For q=1 ordinary statistical mechanics is recovered.
For q different from 1 this is a kind of q-deformed version of statistical mechanics.

A more modern interpretation is that this is an effective description taking into account temperature fluctuations,
or generally the fluctuations of some variance parameter, the so-called superstatistics approach.
The more general formalism is in particular useful for the description of systems with long-range interactions,
multifractal behaviour, and fluctuations of temperature or energy dissipation rate. Examples of physical applications
are nonequilibrium systems with a stationary state (including turbulent flows), scattering processes in elementary particle
physics, and also applications for the dynamics of frequency fluctuations in power grids,
as well as environmental time series such as pollution concentrations.
Look at my most recent publications on the subject in Nature Energy 2018 and Phys. Rev. Research 2019.

Quite a long list of references on nonextensive statistical mechanics and its recent
successful applications can be found here.

Here is a popular science article in the Science magazine (23 August 2002) on this approach.

As an example, the following picture shows differential cross sections of hadronic particles
as produced in e+e- annihilation experiments.
The measurements were done by the TASSO and DELPHI collaboration.
Essentially the figure shows how many particles with a given transverse momentum p_T
are produced at a certain center-of-mass energy E. The solid lines
are analytical predictions of a model based on nonextensive statistical mechanics.
There is excellent agreement between theory and experiments.
This can also be applied to cosmic rays, see recent paper of mine in Scientific Reports (2018).

More details can be found in
C. Beck, Non-extensive statistical mechanics and particle spectra in elementary interactions, Physica 286A , 164 (2000)