The purpose of this special session is to bring together researchers working on various stability issues for coherent structures, special solutions of partial differential equations and their lattices, such as periodic and solitary waves. Many aspects of stability/instability will be discussed, from spectral to nonlinear. It is expected that the speakers will spend some time addressing potential future directions in their respective fields in order to stimulate further discussion and research. The particular emphasis will be given to recent results of Coherent structures in single and multi-component nonlinear wave-type and nonlinear lattice equations with local and/or nonlocal terms as well as orbital and asymptotic stability of breathers and gap solitons, multi-solitons etc This minisymposium touches, via a diverse cohort of experts, upon the current state-of-the-art in this field and the challenges that lie ahead. A balanced perspective encompassing theory, computation and experiment will be sought that should be of value to newcomers, as well as to seasoned researchers in the field.