This book is made up of two essays on the role of time in probability and quantum physics. In the first one, K L Chung explains why, in his view, probability theory starts where random time appears. This idea is illustrated in various probability schemes and the deep impact of those random times on the theory of the stochastic process is shown. In the second essay J-C Zambrini shows why quantum physics is not a regular probabilistic theory, but also why stochastic analysis provides new tools for analyzing further the meaning of Feynman's path integral approach and a number of foundational issues of quantum physics far beyond what is generally considered. The role of the time parameter, in this theory, is critically re-examined and a fresh way to approach the long-standing problem of the quantum time observable is suggested. Contents: Introduction to Random Time: Prologue; Stopping Time; Martingale Stopped; Random Past and Future; Other Times; From First to Last; Gapless Time; Markov Chain in Continuum Time; The Trouble with the Infinite; Introduction to Quantum Randomness: Classical Prologue; Standard Quantum Mechanics; Probabilities in Standard Quantum Mechanics; Feynman's Approach to Quantum Probabilities; Schrödinger's Euclidean Quantum Mechanics; Beyond Feynman's Approach; Time for a Dialogue. Readership: Graduate students in mathematics and physics.