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The study of univalent functions dates back to the early years of the 20th century, and is one of the most popular research areas in complex analysis. This book is directed at introducing and bringing up to date current research in the area of univalent functions, with an emphasis on the important subclasses, thus providing an accessible resource suitable for both beginning and experienced researchers. Contents Univalent Functions - the Elementary Theory Definitions of Major Subclasses Fundamental Lemmas Starlike and Convex Functions Starlike and Convex Functions of Order Strongly Starlike and Convex Functions Alpha-Convex Functions Gamma-Starlike Functions Close-to-Convex Functions Bazilevic Functions B1( ) Bazilevic Functions The Class U( ) Convolutions Meromorphic Univalent Functions Loewner Theory Other Topics Open Problems