Directional solidification of eutectic alloys shows different types of eutectic morphologies. These include lamellar, rod, oscillating and tilting modes. The growth of these morphologies occurs with a macroscopically planar interface. However, under certain conditions, the planar eutectic front becomes unstable and gives rise to a cellular or a dendritic structure. This instability leads to the cellular/dendritic structure of either a primary phase or a two-phase structure. The objective of this work is to develop a fundamental understanding of the instability of eutectic structure into cellular/dendritic structures of a single phase and of two-phases;Experimental studies have been carried out to examine the transition from a planar to two-phase cellular and dendritic structures in a ceramic system of Alumina-Zirconia (Al[subscript]2O[subscript]3-ZrO[subscript]2) and in a transparent organic system of carbon tetrabromide and hexachloroethane (CBr[subscript]4- C[subscript]2Cl[subscript]6). Several aspects of eutectic interface stability have been examined;Studies in the colony structures show that the planar eutectic temperature decreases as the velocity is increased. However, as the planar interface becomes unstable, the temperature of the two-phase cellular structure increases with velocity, goes through a maximum, and finally decreases as the cellular structure transforms into a two-phase dendritic structure. Careful experiments on both transparent as well as ceramic systems have shown deviations of the eutectic spacings in these colony structures from the planar interface model. Based on these results, a new model is proposed to explain the steady-state features of the colony growth;Experimental studies in the instability of eutectics into a single phase formation have shown that the interactions in the microstructures give rise to oscillations between the two stable morphologies around the threshold conditions. The effect of velocity on the oscillation has been examined quantitatively, and it was found that the oscillations decrease as the velocity is increased. A finite band of velocity was observed in which oscillating structures were found to exist. Furthermore, the composition of the alloy showed a large effect on the oscillation of the actual interface velocity which was found to increase as the alloy composition deviated further from the eutectic composition.