This thesis extends earlier works by Wheeler, Boettinger and McFadden [1, 2], and Aziz [9], on a non-equilibrium phenomenon commonly observed in rapid solidification, called 'solute trapping'. The phase-field model by Wheeler, Boettinger and McFadden (denoted as WBM2), and a sharp interface model, known as the Continuous Growth Model (CGM), due to Aziz are found to show similar solute trapping behaviours. Numerical and asymptotic analyses are carried out on the WBM2 model. The numerical results establish the possible cause of solute trapping which is, the relative size of the diffusive length scale of the solute field, and the characteristic thickness of the interface. As the solidification velocity increases, the diffusive length scale of the solute field decreases. When its value becomes comparable or smaller than the characteristic thickness of the interface, solute trapping occurs. This relationship cannot be realised by the Continuous Growth Model because it is a sharp interface model, in which, the interface is assumed to have negligible thickness. This result emphasises the capability of a phase-field model in studying the solute trapping phenomenon. The asymptotic analysis successfully produces an explicit form for the 'diffusive speed', which is an important parameter in solute trapping as it scales the solidification velocity. Solute trapping becomes important when the solidification velocity exceeds this diffusive speed. The explicit expression obtained for the diffusive speed, relates it directly to the material parameters of the alloy; this is the first theory to provide such a relationship. A comparison with values calculated from experimental data on solute trapping shows that this expression supports the experimental results. Another non-equilibrium effect found in rapid solidification processes, is the 'kinetic undercooling' effect. This effect is also successfully captured by the WBM2 model, where the numerical values of the interface temperature is found to decrease with the interface velocity.