Let compact n-dimensional Riemannian manifolds (M,g), (M^, g) a diffeomorphism u\sb0:M→M^, and a constant p>n be given. Then sufficiently small L\spp bounds on the curvature of M^ and on the difference of g and u\sbsp0\* g guarantee that u\sb0 can be continuously deformed to a harmonic diffeomorphism. A vector field v is constructed on the space of mappings u which are L\sp2,p close to u\sb0 by solving the nonlinear elliptic equation Δv+Rc^ v=−Δu. It is shown that under sufficient conditions on u\sb0 and on the curvature Rm^ of the target, the integral curve u\sbt of this vector field converges to a harmonic diffeomorphism. Since the objects we work with, such as v and its derivatives, live in bundles over M, to prove regularity results we must first adapt standard techniques and results of elliptic theory to the bundle case. Among the generalizations we prove are Moser iteration, a Sobolev embedding theorem, and a Calderon-Zygmund inequality.