In this chapter, we study the Cauchy problem for partial differential equations with analytic coefficients, focusing on the linear case but also considering results for non–linear equations. The formulation of the problem and the concept of a characteristic surface do not require analyticity assumptions and are essential for both this chapter and those that follow. Our main result is the Cauchy–Kovalevskaya Theorem, which guarantees the existence and uniqueness of solutions within the class of analytic functions, assuming all data are analytic. We provide a classical proof using majorant functions and discuss alternative approaches. This theorem, though significant, has limitations, prompting further investigation into uniqueness, particularly in the Holmgren Theorem, which relaxes regularity assumptions.