I. PD-Operators with Complex Arguments -- 1. PD-Operators with Constant Analytic Symbols -- 2. Fourier Transformation of Arbitrary Analytic Functions. Complex Fourier Method -- 3. PD-Equations whose Symbols are Formal Series -- II. The Cauchy Problem in the Complex Domain -- 4. Cauchy-Kovalevskaya Theory in Spaces of Analytic Functions with Pole-type Singularities -- 5. Exponential theory of the Cauchy problem -- 6. PD-Operators with Variable Analytic Symbols -- III. PD-Operators with Real Arguments -- 7. Spaces of Test Functions and Distributions -- 8. Analytic PD-operators with Real Arguments. Applications -- References -- Author Index -- Index of Basic Formulae.