  We study covariant differential calculus on the quantum spheres S_q^2N-1.
  Two classification results for covariant first order differential calculi are
proved. As an important step towards a description of the noncommutative
geometry of the quantum spheres, a framework of covariant differential calculus
is established, including a particular first order calculus obtained by
factorization, higher order calculi and a symmetry concept.
