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In this paper we develop a new approach to the theory of Fourier integral operators. It allows us to represent the Schwartz kernel of a Fourier integral operator by one oscillatory integral with a complex phase function. We consider Fourier integral operators associated with canonical transformations, having in mind applications to hyperbolic equations. As a by-product we obtain yet another formula for the Maslov index. c©1994 John Wiley & Sons, Inc. Let M be a C∞-manifold without boundary, dimM = n, and T ∗M\0 be the cotangent bundle without the zero section. We consider the Lagrangian manifold Λ ⊂ (T ∗M\0) × (T ∗M\0