Something about the author Dr. Shizan Fang(bom in 1963 ). Professor of University of Burgundy(Dijon. FranceS, obtained his PhD degree at University of Paris VI in February 1990 and then worked there as "maitre de Conferences" during 1990-1996. His main interests of research are in the field of "Analysis. Geometry and Probability". He has published some first rate results on the subjects "Geometric Analysis on the Wiener Space". "Geometric Stochastic Analysis on Riemannian Path Spaces and Loop Groups". "Stochastic Differential Equations and Flow of Homeomorphism". Abstract of the book Malliavin Calculus is the theory of infinite dimensional differential calculus, which is suitable for functionals involved in diffusion theory, stochastic control, financial market models, etc. It also provides infinite dimensional examples in Dirichlet forms theory, in Functional Inequalities Analysis, etc. The main purpose of this book is to give a foundation of Malliavin Calculus, as well as some insights toward further researches in the field of path and loop spaces.