Atmospheric aerosol particles influence our planet's climate and contribute to poor air quality, increased mortality and degraded visibility. Central to these issues is how atmospheric aerosol particles interact with gas species to affect chemistry and cloud formation. Recent research shows that some aqueous solutions relevant to atmospheric aerosol (notably secondary organic material, which constitutes a large mass fraction of atmospheric aerosol particles) can be highly viscous and can behave mechanically like a solid. This has led to suggestions that these particles exist out of equilibrium with the gas phase in the atmosphere, with implications for heterogeneous chemistry and ice nucleation. In order to quantify any kinetic limitations, it is vital to have quantitative data about the diffusion of various relevant species within these materials. This thesis describes the direct measurement and application of water diffusion coefficients in aqueous solutions relevant to atmospheric aerosol, including sucrose and secondary organic material. A water diffusion model is developed, validated and used with a new parameterisation of the water diffusion coefficient in secondary organic material to quantify the rate of uptake and loss of water from aerosol particles. It is shown that, although this material can behave mechanically like a solid, at 280 K water diffusion is not kinetically limited on timescales of 1 s for atmospheric-sized particles. This is not the case, however, for colder conditions: modelling of 100 nm particles predicts that under mid to upper tropospheric conditions radial inhomogeneities in water content produce a low viscosity surface region and more solid interior. This may significantly affect aerosol chemistry and the ability of particles to nucleate ice. Also reported are the diffusion coefficients of sucrose in aqueous sucrose at higher concentrations than have been previously investigated. These measurements provide insights into the role of organic molecules in aerosol evaporation and chemistry. Together with the diffusion coefficients of water measured in this material, they will also offer a valuable means to study the fundamental nature of diffusion in a simple but widely used material, and specifically the breakdown of the Stokes-Einstein relationship.