Lectures on Probability Theory and Statistics : Ecole d’Eté de Probabilités de Saint Flour XXVIII 1998 Variétés vecteurs covecteurs diffuseurs codiffuseurs Semimartingales dans une variété et géométrie d’ordre 2 Connexions et martingales Fonctions convexes et comportement des martingales Mouvements browniens et applications harmoniques Preface Estimating regression functions from Hölder balls Estimating regression functions from Sobolev balls Spatial adaptive estimation on Sobolev balls Estimating signals satisfying differential inequalities Aggregation of estimates I Aggregation of estimates II Estimating functionals I Estimating functionals II Noncommutative probability and operator algebra background Addition of freely independent noncommutative random variables Multiplication of freely independent noncommutative random variables Generalized canonical form noncrossing partitions Free independence with amalgamation Some basic free processes Random matrices in the large N limit Free entropy. This volume contains lectures given at the Saint Flour Summer School of Probability Theory during 17th Aug. 3rd Sept. 1998. The contents of the three courses are the following: Continuous martingales on differential manifolds. Topics in non parametric statistics. Free probability theory. The reader is expected to have a graduate level in probability theory and statistics. This book is of interest to PhD students in probability and statistics or operators theory as well as for researchers in all these fields. The series of lecture notes from the Saint Flour Probability Summer School can be considered as an encyclopedia of probability theory and related fields.