The approach to the Cauchy problem taken here by the authors is based on the use of Fourier integral operators with a complex-valued phase function, which is a time function chosen suitably according to the geometry of the multiple characteristics. The correctness of the Cauchy problem in the Gevrey classes for operators with hyperbolic principal part is shown in the first section of the text. In the second section, the correctness of the Cauchy problem for effectively hyperbolic operators is proved with a precise estimate of the loss derivatives. This method can be applied to other (non) hyperbolic problems. The text is based on a course of lectures given for graduate students but will be of interest to researchers interested in hyperbolic partial differential equations. In the latter part, the reader is expected to be familiar with the theory of pseudo-differential operators.