The theme of this book is the philosophy that any particle flow generates a particle dynamics, in a suitable geometrical framework. It introduces the reader in a gradual and accessible manner to this subject, covering topics that include: geometrical and physical vector fields; field lines; flows; stability of equilibrium points; potential systems and catastrophe geometry; field hypersurfaces; bifurcations; distribution orthogonal to a vector field; extrema with nonholonomic constraints; thermodynamic systems; energies; geometric dynamics induced by a vector field; magnetic fields around piecewise rectilinear electric circuits; geometric magnetic dynamics; and granular materials and their mechanical behavior. Primary audience: First-year graduate students in mathematics, mechanics, physics, engineering, biology, chemistry, economics. Part of the book can be used for undergraduate students. Secondary audience: The book is addressed also to professors and researchers whose work involves mathematics, mechanics, physics, engineering, biology, chemistry, and economics.