Secondary calculus is the result of a natural evolution of the classical geometrical theory of partial differential equations (PDE) originated by Sophus Lie. In particular, it allows the construction of a general theory of PDE, in the same manner as algebraic geometry does with respect to algebraic equations. There are strong indications that secondary calculus may become a natural language for quantum field theory, just in the same way as standard calculus is for classical physics. From the mathematical point of view, secondary calculus is a complex mathematical construction putting into a natural interrelation many parts of modern mathematics such as commutative and homological algebra, algebraic and differential topology, differential geometry, etc. The strategic goal of the Diffiety Schools to involve interested participants into a series of large scale research programs the Levi-Civita Institute is launching.