This volume shares and makes accessible new research lines and recent results in several branches of theoretical and mathematical physics, among them Quantum Optics, Coherent States, Integrable Systems, SUSY Quantum Mechanics, and Mathematical Methods in Physics. In addition to a selection of the contributions presented at the "6th International Workshop on New Challenges in Quantum Mechanics: Integrability and Supersymmetry", held in Valladolid, Spain, 27-30 June 2017, several high quality contributions from other authors are also included. The conference gathered 60 participants from many countries working in different fields of Theoretical Physics, and was dedicated to Prof. Véronique Hussin¿an internationally recognized expert in many branches of Mathematical Physics who has been making remarkable contributions to this field since the 1980s. The reader will find interesting reviews on the main topics from internationally recognized experts in each field, as well as other original contributions, all of which deal with recent applications or discoveries in the aforementioned areas.

Chapter1. Curvature as an integrable deformation.- Chapter2. Trends in supersymmetric quantum mechanics.- Chapter3. Coherent states in Quantum Optics: An oriented overview.- Chapter4. Higher Order Quantum Superintegrability: a new "Painlevé conjecture.- Chapter5. Supersymmetries in Schrodinger - Pauli equationsand in Schrodinger equations with position dependent mass.- Chapter6. Nonlinear supersymmetry as a hidden symmetry.- Chapter7. Coherent and squeezed states: introductory review of basic notions, properties and generalizations.- Chapter8. Trace Formulas Applied to the Riemann -Function.- Chapter9. "Real- and complex-energy non-conserving particle

number pairing solution".- Chapter10. Jacobi polynomials as su(2; 2) unitary irreducible representation.- Chapter11. Innite square-well, trigonometric Poschl-Teller and other potential wells with a moving barrier.- Chapter12. Variational Method Applied to Schrodinger-like Equation.- Chapter13. The Lippmann-Schwinger formula and one dimensional models with Dirac delta interactions.- Chapter14. Hermite coherent states for quadratic refractive index optical media.- Chapter15. Analysis of CPN¿¿¿¿1 sigma models via soliton surfaces.- Chapter16. On the equivalence between Type I Liouville dynamical systems in the plane and the sphere.- Chapter17. Construction of partial dierential equations with conditional symmetries.- Chapter18. An Integro-dierential Equation of the Fractional Form: Cauchy Problem and Solution.- Chapter19. Quasi-integrability and some aspects of SU(3) Toda eld theory.- Chapter20. On some aspects of unitary evolution generated by non-Hermitian Hamiltonians.