Rapid solidification of metals and alloys has developed from an insignificant activity into an industrial practice and an important process for research studies. It is generally brought about by one of the following two methods. The first is by rapid quenching from the melt, e.g. melt spinning. The second method involves undercooling the liquid by a large degree below its melting temperature prior to nucleation. In this way, a large driving force for rapid solidification accumulates due to the large difference in Gibbs free energy between the solid and the liquid state. This book presents on overview of solidifcation and microstructural evolution of deeply undercooled metallic melts. The experimental methods for acheiving large degrees of undercooling prior to nucleation are also considered. Detailed examples are provided for the pure Cu, Cu-O and Cu-Sn alloy systems. Special emphasis is placed on issues including grain refinement, twinned dendritic growth and dendrite- tip splitting, as these are currently one of the most widely studied and yet not fully furnished microstructural effects observed during solidification far from equilibrium.