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 Mémoires de la Société Mathématique de France 2011; 136 pp; softcover Number: 123 ISBN-10: 2-85629-304-2 ISBN-13: 978-2-85629-304-1 List Price: US$42 Individual Members: US$37.80 Order Code: SMFMEM/123 This work consists two parts. In the first part, the author studies completely the heat equation method of Menikoff-Sjöstrand and applies it to the Kohn Laplacian defined on a compact orientable connected CR manifold. He then gets the full asymptotic expansion of the Szegő projection for $$(0, q)$$ forms when the Levi form is non-degenerate. This generalizes a result of Boutet de Monvel and Sjöstrand for $$(0,0)$$ forms. The author's main tools are Fourier integral operators with complex valued phase Melin and Sjöstrand functions. In the second part, the author obtains the full asymptotic expansion of the Bergman projection for $$(0, q)$$ forms when the Levi form is non-degenerate. This also generalizes a result of Boutet de Monvel and Sjöstrand for $$(0,0)$$ forms. He introduces a new operator analogous to the Kohn Laplacian defined on the boundary of a domain and applies the heat equation method of Menikoff and Sjöstrand to this operator. He obtains a description of a new Szegő projection up to smoothing operators and gets his main result by using the Poisson operator. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in analysis. Table of Contents Introduction Part I. On the singularities of the Szegö projection for $$(0, q)$$ forms Part II. On the singularities of the Bergman projection for $$(0, q$$ forms Bibliography