Roles of Plurisubharmonic Functions

We recall the basics of multiplier ideal sheaves and formulate our recent solution of Demailly's strong openness conjecture on multiplier ideal sheaves and related results. Then we present some applications in complex geometry, including some new results related to the vanishing and finiteness theorems for analytic cohomology groups with multiplier ideal sheaves in the case of pseudo-effective line bundles over holomorphically convex manifolds, and to the generalized Siu lemma and pseudo-effectiveness of the twisted relative pluricanonical bundles and their direct images.