There has been quite a lot of literature on the applications of $q$-numbers, $q$-derivatives, $q$-deformations, etc, of various algebraic models of physics. Such applications range from $q$-deformations of simple harmonic oscillator(s) and angular momentum algebras to the development of quantum groups and their applications in nuclear physics, particle physics and field theories. They can be -roughly- divided in two broad categories (although experts might argue that such classifications can be made much more fine): - *Applications of phenomenological nature:* Various $q$-deformed oscillators or $q$-deformed rotators have been used. Their spectrums, transition rates, matrix elements etc are proved to depend on the deformation parameter(s). "Playing" with the parameters allows curve fitting to experimental data. In lots of cases deformations based on more than one parameters (i.e. $q,p,..$-deformations) have been used. An interesting characteristic of such applications is that in many cases -for example in the fitting of the vibrational and rotational spectra of nuclei and molecules- the number of phenomenological $q$-parameters needed is significantly fewer than the number of traditional phenomenological parameters required to fit the same spectral data. (In many cases, the physical interpretation of such deformation parameters still [lingers -see p.179-][1] and imposes ideas of a more fundamental nature). In modern times, such ideas are also finding applications in semi-phenomenological cosmological models. See for example [9][2] where the dark energy is essentialy considered to be a $q$-deformed scalar field. Just to mention a few papers in this category (my former phd advisor has quite some work on the field): 1. [Generalized deformed oscillator and nonlinear algebras, C Daskaloyannis 1991 J. Phys. A: Math. Gen. 24 L789][3] 2. [Coupled $Q$-oscillators as a model for vibrations of polyatomic molecules, D.Bonatsos, C.Daskaloyannis, The Journal of Chemical Physics 106, 605 (1997)][4] 3. [Quantum groups and their applications in nuclear physics, D.Bonatsos, C.Daskaloyannis, Progress in Particle and Nuclear Physics, v.43, 1999, p. 537-618][5] (see also [here][6] for the arxiv version). 4. [The many-body problem for $q$-oscillators, E G Floratos, Journal of Physics A: Mathematical and General, Volume 24, Number 20, 1991 ][7] 5. [Dynamical algebra of the $q$‐deformed three‐dimensional oscillator, J. Van der Jeugt, J. of Math. Phys. 34, 1799 (1993)][8] 6. [WKB equivalent potentials for the $q$-deformed harmonic oscillator, D Bonatsos, C Daskaloyannis and K Kokkotas, J. of Phys. A: Math. and Gen., Volume 24, Number 15, 1991][9] 7. Introduction to Quantum Algebras, Maurice R. Kibler, [arXiv:hep-th/9409012][10] (see especially the discussion in sections 7,8 and the references). 8. An Introduction to Quantum Algebras and Their Applications, R. Jaganathan, [arXiv:math-ph/0003018][11] (see the discussion of p.11-13) 9. [Interacting Dark Matter and $q$-Deformed Dark Energy Nonminimally Coupled to Gravity, Emre Dil, Advances in High Energy Physics, Volume 2016 (2016), Article ID 7380372][2] - *Applications of a more conceptual or fundamental nature:* -->Such as for example the introduction and study of non-commutative space-times. Here, the non-commutativity is frequently expressed through deformed commutation relations between the coordinates or the functions on the space-time manifolds, leading to deformed algebras and non-commutative geometries. Fundamental quantization problems are frequently discussed in this setting: See for example [this paper][12], [this paper][13] and [this one][14] (among lots of works in a similar spirit). The [relation of $q$-mathematics and quantum groups to deformation quantization][15] is another hot topic with a significant number -imo- of open questions. In [Minimal areas from q-deformed oscillator algebras][16] the authors argue that non-commutative space-times with dynamical commutation relations between the coordinates imply $q$-deformed algebras of observables and a kind of converse argument is also supplied. -->[The quantum inverse scattering method][17] and the definition(s) and study of [quantum integrability (see p. 269)][18] have given rise to [quantum groups][19] and [quantum algebras][20]. The mathematical developments associated with them have been greatly inspired and have -in return- significantly contributed to such directions of study. [Jule Lamers'][21] answer provides further details and references on that point. [S. Majid's book][22] is also worth to be mentioned as it contains a cataclysm of such ideas and applications. -->The case of [deformed particles][23] (deformed bosons or fermions) which [interpolate between different statistics][24] has been another line of interesting applications of this kind. A significant amount of related literature can be found at mathematical physics journals such as the Journal of Mathematical Physics, Journal of Physics A: Mathematical and General, Communications of Mathematical Physics, [SIGMA][25] etc. [1]: https://books.google.gr/books?id=SD5hDQAAQBAJ&pg=PA179&lpg=PA179&dq=quantum%20inverse%20scattering%20method%20q-deform&source=bl&ots=1bonS11gDJ&sig=drS1gQPH7vnpIeLxi9o1mbbQ2LM&hl=el&sa=X&ved=0ahUKEwjfw9nlsvXVAhWJ6xoKHVCUBjEQ6AEISzAF#v=onepage&q=quantum%20inverse%20scattering%20method%20q-deform&f=false [2]: https://www.hindawi.com/journals/ahep/2016/7380372/ [3]: http://iopscience.iop.org/article/10.1088/0305-4470/24/15/001 [4]: http://aip.scitation.org/doi/citedby/10.1063/1.473189 [5]: http://www.sciencedirect.com/science/article/pii/S0146641099001003?via%3Dihub [6]: https://arxiv.org/pdf/nucl-th/9909003.pdf [7]: http://iopscience.iop.org/article/10.1088/0305-4470/24/20/009/meta [8]: http://aip.scitation.org/doi/abs/10.1063/1.530138 [9]: http://iopscience.iop.org/article/10.1088/0305-4470/24/15/002/meta [10]: https://arxiv.org/abs/hep-th/9409012 [11]: https://arxiv.org/abs/math-ph/0003018v1 [12]: https://cds.cern.ch/record/369890/files/9811027.pdf [13]: http://www.sciencedirect.com/science/article/pii/S0393044097000065?via%3Dihub [14]: https://link.springer.com/chapter/10.1007%2F978-3-7643-8736-5_15 [15]: http://aip.scitation.org/doi/abs/10.1063/1.1786681 [16]: http://iopscience.iop.org/article/10.1088/1751-8113/43/42/425202/meta [17]: https://en.wikipedia.org/wiki/Quantum_inverse_scattering_method [18]: https://books.google.gr/books?id=0oF6h4eT-50C&pg=PA269&lpg=PA269&dq=quantum%20inverse%20scattering%20method%20q-deform&source=bl&ots=sVKzqx9Gjr&sig=S3tX15pP-nTZPLcAplHBr-oAljE&hl=el&sa=X&ved=0ahUKEwjfw9nlsvXVAhWJ6xoKHVCUBjEQ6AEIXTAI#v=onepage&q=quantum%20inverse%20scattering%20method%20q-deform&f=false [19]: https://en.wikipedia.org/wiki/Quantum_group [20]: https://arxiv.org/pdf/hep-th/9409012.pdf [21]: https://mathoverflow.net/a/279309/85967 [22]: https://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521460323 [23]: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.43.4111 [24]: http://iopscience.iop.org/article/10.1088/0305-4470/26/16/018 [25]: https://www.emis.de/journals/SIGMA/