L² estimates for the (Formula Presented.) operator

We present the theory of twisted L² estimates for the Cauchy–Riemann operator and give a number of recent applications of these estimates. Among the applications: extension theorem of Ohsawa–Takegoshi type, size estimates on the Bergman kernel, quantitative information on the classical invariant metrics of Kobayshi, Caratheodory, and Bergman, and sub elliptic estimates on the (Formula Presented.)-Neumann problem. We endeavor to explain the flexibility inherent to the twisted method, through examples and new computations, in order to suggest further applications.