This dissertation consists of two parts, connected by the overarching theme of the dynamics of structured waves with internal degrees of freedom. Part I concerns light, whose internal degree of freedom is polarisation. We investigate the helicity, or handedness of light, which is a good quantum number for massless fields in general and light in particular. In free space it is always possible to describe the light field in a basis left- and right handed helicity modes which are solutions of Maxwell's equations, regardless what spatial structure is chosen. This is useful for bases of highly inhomogeneous waves, such as Bessel waves, for which the spin cannot be unambiguously defined. In chapter 1 we study the conservation of helicity and the preservation of its underlying symmetry, electric-magnetic duality symmetry when light travels through inhomogeneous and/or anisotropic media. We will discuss some of the unique properties of duality symmetric media and reformulate Maxwell's equations in such a way that the decoupling of different helicities for duality symmetric media becomes apparent. The feasibility of constructing duality symmetric media is discussed at the end of the chapter. In chapter 2 we consider superpositions of plane electromagnetic waves in free space. Such superpositions typically interfere. We present superpositions of up to six plane waves which defy this expectation by having a perfectly homogeneous mean square of the electric field. Because most matter interacts much stronger with the electric than with the magnetic field, these superpositions can be considered noninterfering. Our superpositions show complex patterns in their helicity densities, of which we will show many examples. We study the effects on our helicity patterns of imperfections that may occur in a realistic experiment: deviations from the optimal amplitudes, phases and polarisations of the superposed waves, small misalignments and partially coherent light. Our superpositions can be used to write chiral patterns in light sensitive liquid crystals. Conversely, these liquid crystals can be used for an `optical helicity camera' which records spatial variations in helicity. In the final paragraph of chapter 2 we discuss some mathematical questions concerning noninterfering superpositions. Part II concerns electrons, whose internal degree of freedom is spin. In chapter 3 we will present analytical solutions of the Dirac equation for an electron vortex beam in a homogeneous magnetic field. Including spin from the beginning reveals that spin polarised electron vortex beams have a complicated azimuthal current structure, containing small rings of counterrotating current between rings of stronger corotating current. Contrary to many other problems in relativistic quantum mechanics, there exist vortex beam solutions with exactly zero spin-orbit mixing in the highly relativistic and nonparaxial regime. Chapter 4 treats the interaction between electron vortex states in a homogeneous magnetic field and light, where we expand and quantise the radiation field in a basis of Bessel modes with definite helicity. Our results apply for magnetic field strength beyond the critical field strength at which the spin contributes as much to the electron's energy as its rest mass. We are able to compute spin flip rates for low lying states, finding a much higher degree of equilibrium spin polarisation than approximations for high lying electron states suggested.