I propose to use computer simulations to predict the thermodynamic stability and kinetics of formation of three-dimensional structures of DNA-linked colloids. I then aim to go beyond simple binary structures and use simulation to explore novel strategies to build multi-component three-dimensional colloidal structures. At present, the complexity of self-assembled colloidal crystals is limited: ordered structures with more than two distinct components are rare. To make more complex structures, particles should bind selectively to their designated neighbours. This may be achieved by coating colloids with single-stranded DNA that hybridises selectively with the complementary sequence on another colloid. However, there are many practical obstacles to go from there to the self assembly of multi-component structures. In order to make progress, we need to understand the factors that determine the thermodynamic stability and, even more importantly, the kinetics of formation of complex structures. Such a numerical study will require a wide range of numerical techniques, many of which do not yet exist. As I have played a key role in the development of the numerical methods to study both the stability and the kinetics of formation of simple colloidal crystals, I am well positioned to make a breakthrough that should have important implications for experimental work in this field. My research will focus on DNA-linked colloidal systems, as this is an active area of experimental research. However, I stress that many of the techniques that I aim to develop are general. During the project, I aim to study the factors that influence the equilibrium phase diagram and the kinetics of passive and active self-assembly of (multi-component) DNA-colloid systems During the project, I aim to study the factors that influence the equilibrium phase diagram and the kinetics of passive and active self-assembly of (multi-component) DNA-colloid systems