We give new sufficient conditions for the integrability and unique
integrability of continuous tangent sub-bundles on manifolds of arbitrary
dimension, generalizing Frobenius' classical Theorem for C^1 sub-bundles. Using
these conditions we derive new criteria for uniqueness of solutions to ODE's
and PDE's and for the integrability of invariant bundles in dynamical systems.
In particular we give a novel proof of the Stable Manifold Theorem and prove
some integrability results for dynamically defined dominated splittings.