This chapter introduces the Cauchy problem for partial differential equations, starting with first-order equations in a real context. It serves two purposes: to familiarize readers with the Cauchy problem and to prepare key concepts for later chapters. The first-order Cauchy problem acts as an intermediate case between ordinary and more general partial differential equations. A brief review of ordinary differential equations is provided for convenience. We focus on the method of characteristic lines for linear equations and give a brief overview of quasilinear equations and fully nonlinear equations.