We study quivers with relations given by non-commutative analogs of Jacobian
ideals in the complete path algebra. This framework allows us to give a
representation-theoretic interpretation of quiver mutations at arbitrary
vertices. This gives a far-reaching generalization of
Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work
come from several sources: superpotentials in physics, Calabi-Yau algebras,
cluster algebras.