This book is organized into nine chapters, the first six of which are on expansion of filtration formulae, Burkholder-Gundy inequalities up to any random time, martingales which vanish on the zero set of Brownian motion, the Azema-Emery martingales and chaos representation, the filtration of truncated Brownian motion, and attempts to characterize the Brownian filtration. The three remaining chapters discuss principle value diffusion times, probabilistic representations of the Riemann zeta function, and progress made on some topics covered in part one. Most of the contents of this text are the subjects of active research, centred on real-value martingales and Brownian mothion, and this volume may be of interest to researchers in probability theory or in more applied fields, such as mathematical finance.