In this paper we construct, for an arbitrary principal bundle (with group a real Lie group) over a differentiable manifold, a sheaf of germs of linear differential operators of first order operating on the sheaf of germs of sections of any differentiable vector bundle associated with the given principal bundle. The commutators of these operators define a structure of Lie algebra (over the real numbers) on the operator sheaf. By considering the sheaves of germs of differential forms on the manifold with values in appropriate vector bundles, we obtain a graded sheaf of operators carrying a structure of graded Lie algebra (over the real numbers). In many respects, the present work represents a generalization of the paper of Frolicher and Nijenhuis [2] which characterized the derivations (of all degrees) of the exterior algebra of real-valued differential forms on a differentiable manifold. The case considered by these authors is obtained by choosing the associated vector bundle to be the product bundle with fibre the real numbers. In addition, the usual theory of covariant differentiation, corresponding to a connection in the given principal bundle, appears as a special case. It is hoped that the theory developed here will also have applications in the systematic study of deformation of the structures defined on a differentiable manifold by continuous pseudogroups of transformations. 1. Fundamental sequence of vector bundles. Let M be a differentiable (i.e., C*) manifold and let P-4M be a differentiable principal bundle with group G, where G is a real Lie group. Let T(M), T(P), and T(G) denote the bundle spaces of the tangent bundles to M, P, and G respectively. Then T(G) is also a Lie group, and T(P)--T(M) is a principal bundle with group T(G) [3 ]. Since G is a subgroup of T(G), we may form the quotient bundle T(P)/G --T(M), and we have