In this talk, we first review the basics of minimal submanifolds and their stability. The concepts of index and nullity, first introduced by J. Simons, provide a deeper understanding of the geometric properties of submanifolds. We focus on the details of computing the index and nullity for minimal submanifolds of some complex Einstein spaces. Then, we analyze the stability of these minimal submanifolds and suggest a criterion for instability in certain cases. (Based on a joint work with M. Kalafat & M. Taşdemir)