Theoretical physicists have predicted that the scaling limits of many twodimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely by using one wellknown tool, Brownian motion, and a new construction, the SchrammLoewner evolution (SLE). This book is an introduction to the conformally invariant processes that appear as scaling limits. The following topics are covered: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; the SchrammLoewner evolution (SLE), which is a Loewner chain with a Brownian motion input; and applications to intersection exponents for Brownian motion. The prerequisites are firstyear graduate courses in real analysis, complex analysis, and probability. The book is suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics. Readership Graduate students and research mathematicians interested in random processes and their applications in theoretical physics. Reviews "This nice book celebrates the fruitful marriage of Brownian motion and complex analysis."  Zentralblatt MATH "This book gives a nice and systematic introduction to the contiuous time conformally invariant processes in the plane, assuming only knowledge of first year graduate real analysis, complex analysis and probability theory ... This books is very well written, and can also be used as a graudate textbook for a topic course on SLE."  Mathematical Reviews Table of Contents  Some discrete processes
 Stochastic calculus
 Complex Brownian motion
 Conformal mappings
 Loewner differential equation
 Brownian measures on paths
 SchrammLoewner evolution
 More results about SLE
 Brownian intersection exponent
 Restriction measures
 Hausdorff dimension
 Hypergeometric functions
 Reflecting Brownian motion
 Bibliography
 Index
 Index of symbols
