Painlevé's work in second order nonlinear ordinary differential equations included finding that the location of the possible branch points and essential singularities of their solutions does not depend on local conditions. This applies to only six equations, the well-known Painlevé I-VI. However few, these equations are now applied to statistical physics, fluid mechanics, random matrices and orthogonal polynomials, and the solutions of the equations play a significant role in nonlinear mathematical physics. In the 16 chapters here the authors emphasize the connection of the Riemann-Hilbert method the classical monodromy theory of linear equations as well as to modern theories of integrable systems. They cover isomonodromy methods and special functions, including the Backlund transformations, describe a case study for asymptotics of the Painlevé II transcendent, including the canonical six-rays, and examine the asymptotics of the Painlevé III transcendent, including the Sine-Gordon reduction and canonical four-rays. Annotation ©2006 Book News, Inc., Portland, OR (booknews.com)