Feynman categories are a new universal categorical framework for generalizing operads, modular operads and twisted modular operads. The latter two appear prominently in Gromov-Witten theory and in string field theory respectively. Feynman categories can also handle new structures which come from different versions of moduli spaces with different markings or decorations, e.g. open/closed versions or those appearing in homological mirror symmetry. For any such Feynman category there is an associated Feynman category of universal operations. These give rise to Gerstenhaber's famous bracket, the pre-Lie structure of string topology, as well as to the Lie bracket underlying the three geometries of Kontsevich built from symplectic vector spaces. As time permits, we will also briefly discuss bar, co-bar and Feynman transforms and how these give rise to master equations, such as the Maurer-Cartan equation or the BV master equation.