The stability mechanisms of a curved front and the selection of pattern formation in dendrite growth and viscous fingering have been long-standing fundamental subjects in the areas of condensed matter physics, material science, crystal growth and fluid mechanics. This book studies the interfacial stability and pattern formation in dynamic systems away from the equilibrium state. In particular, it deals with the two prominent prototype systems: dendrite growth in solidification and viscous fingering in Hele--Shaw flow. It elucidates the key problems step by step and systematically derives their mathematical solutions on the basis of the newly established interfacial wave theory. Finally, it carefully examines these results by comparisons with the available experimental results.