The homological and algebraic behaviour of powers of an ideal has been a subject of several research papers in recent years. In this series of talks, first I will discuss the well known results about the stability of the set of associated prime ideals and the depth function of powers of an ideal in a Noetherian ring. Later, based on my joint work with Juergen Herzog and Takayuki Hibi, I will discuss the concept of strong persistence property of powers of an ideal and polarization of Koszul cycles and then use it to study the index of stability for the set of associated prime ideals and the depth function of powers of an ideal.