This volume is an expansion of lectures given by the author at the Park City Mathematics Institute (Utah) in 2008, and on other occasions. The purpose of this volume is to describe analytic techniques useful in the study of questions pertaining to linear series, multiplier ideals, and vanishing theorems for algebraic vector bundles. The author aims to be concise in his exposition, assuming that the reader is already somewhat acquainted with the basic concepts of sheaf theory, homological algebra, and complex differential geometry. In the final chapters, some very recent questions and open problems are addressed--such as results related to the finiteness of the canonical ring and the abundance conjecture, and results describing the geometric structure of Kahler varieties and their positive cones.