The mathematics in this volume of the Encyclopaedia, the third in the subseries on PDEs, is more advanced than that in EMS 30 (PDE I). The reader should already be familiar with what is found in that volume. EMS 32 covers in detail two very important topics in PDE: the Cauchy problem and the qualitative theory of second order linear equations. It should be of interest to advanced students and researchers in mathematics and to scientists in neighboring fields with strong mathematical backgrounds.