This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings:

1. ODE theory;
2. Maximum Principles (weak, strong and propagation principles);
3. Lie groups (with an emphasis on the construction of Lie groups).

This book also provides an introduction to the basic theory of Geometrical Analysis, with a new foundational presentation based on Ordinary Differential Equation techniques, in a unitary and self-contained way.