This open access book published in the Soft and Biological Matter series presents an introduction to selected research topics in the broad field of flowing matter including the dynamics of fluids with a complex internal structure from nematic fluids to soft glasses as well as active matter and turbulent phenomena. Flowing matter is a subject at the crossroads between physics mathematics chemistry engineering biology and earth sciences and relies on a multidisciplinary approach to describe the emergence of the macroscopic behaviours in a system from the coordinated dynamics of its microscopic constituents. Depending on the microscopic interactions an assembly of molecules or of mesoscopic particles can flow like a simple Newtonian fluid deform elastically like a solid or behave in a complex manner.