The scale-invariant superspace formulation of d = 10 Einstein-Yang-Mills system is considered which gives a most economical and clear presentation of the full on-shell content of the theory. Discussion of supersymmetric higher order corrections to the effective action of string theory is based on the corresponding supergeometry. A geometrical formulation of the heterotic string theory in superspace with 10 + 496 bosonic and 16 fermionic coordinates is presented. Minimal constraints on this superspace are found which provide the local fermionic and two types of bosonic world-sheet symmetries of the heterotic string. The relation between this superspace and the anomaly-free d = 10 Einstein-Yang-Mills supergravity is discussed. The existence of a self-dual world-sheet torsion, which is required by the world-sheet symmetries, explains the geometrical origin of the absence of right movers in the gauge degrees of freedom of the heterotic string. The action of the heterotic string in the full curved background has a particularly simple form: .