This book aims to predict the Mathematical Foundations of Statistical Mechanics and explain the measurable properties of macroscopic systems on the basis of the properties and behaviour of the microscopic constituents of those systems. Entropy of an isolated system may increase spontaneously into the future but not into the past. But the dynamical laws governing the motion of the micro-constituents are, at least on the standard views of those laws as being the usual laws of classical or quantum dynamics, time reversal invariant. Introducing probabilistic elements into the underlying theory still does not by itself explain where time asymmetry gets into the explanatory account.