In this talk, we mainly discuss extension theorems with singular weights on Kähler manifolds and their applications. We will discuss a refined optimal L² extension theorem with singular weights on Kähler manifolds and its applications to twisted pluricanonical sheaves. We will also discuss an extension theorem with singular weights for cohomology classes defined on a not necessarily reduced analytic subvariety and its applications to injectivity theorems and vanishing theorems. These are joint works with Professor Xiangyu Zhou.