The “current algebras” of elementary particle physics and quantum field theory are interpreted as infinite dimensional Lie algebras of a certain definite kind. The possibilities of algebraic structure and certain types of representations of these algebras by differential operators on manifolds are investigated, in a tentative way. The Sugawara model is used as a typical example. A general differential geometric method (involving jet spaces) for defining currents associated with classical field theories is presented. In connection with the abstract definition of current algebras as modules, a purely module-theoretic definition of a “differential operator” is presented and its properties are studied.