Time delays are present in control systems (sensing,actuation and decision-making) or they are inherentparts of dynamical systems (chatter in metalmachining, heredity in biology). Delays are known tocause detrimental effects to system behavior, butthey also favor system performance, assistcontroller's success and even recover stability.This counter-intuitive and intriguing phenomenon isa strong motivation for engineering, mathematics,biology and physics disciplines. This book addressesone of the timeliest and the most crucial problemsstudied since 1960s: the asymptotic stability oftime delay systems. The main focus is on a novelparadigm, Cluster Treatment of Characteristic Rootsmethodology, which can treat the stability of lineartime invariant (LTI) control systems with multipledelays. The author hopes that the methodology willnot only help various disciplines to investigatesimilar problems, but it will also inspire newresearch directions in algebraic geometry, numericaltechniques, stability of multinomials, mappingbetween finite and infinite dimensional systems,robustness and control design.