This book is based on lectures presented by the author at DePaul University in July 1986. The lectures cover three main topics. The first is local time theory for Brownian motion and some geometrical inequalities for harmonic functions in the upper half-plane \(R^{n+1}_+\). The author sketches a proof of the inequalities obtained by Barlow and Yor for the maximal local time functional. The second topic concerns a probabilistic treatment of Riesz transforms in \(R^{n+1}_+\), and semimartingale inequalities. The author proves semimartingale inequalities of the type usually obtained for martingales. The final topic centers on a discussion of the Ornstein-Uhlenbeck semigroup and P. A. Meyer's extension of the Riesz inequalities for the infinite-dimensional version of this semigroup. One of the major results of the book is the establishment of inequalities for the density of the area integral.