In this review we investigate the mathematical description of the distortion of clearly recognisable structures in phenomenological physics. The coherent structures we will explicitly deal with are surface waves on a layer of fluid, kink transitions in magnetic material, plane vortices, swirling flows in cylindrical pipes and periodic patterns in pattern formation equations. The deformation of such structures will be studied for perturbations of different kinds. Problems with dissipation as a perturbation include the decay of surface waves under the influence of uniform damping and viscosity, and the viscous decay of vortices along branches that connect to a Leith vortex. Inhomogeneity as a perturbative effect will be studied for waves above slowly varying topography, for the particle description of kinks in inhomogeneous magnetic materials and for swirling flows in slowly expanding pipes. Finally, slow variations in pattern formation equations will result in phase-diffusion or amplitude equations.