  We study covariant differential calculus on the quantum spheres S_q^{N-1}
which are quantum homogeneous spaces with coactions of the quantum groups
O_q(N).
  The first part of the paper is devoted to first order differential calculus.
A classification result is proved which says that for N>=6 there exist exactly
two covariant first order differential calculi on S_q^{N-1} which satisfy the
classification constraint that the bimodule of one-forms is generated as a free
left module by the differentials of the generators of S_q^{N-1}. Both calculi
exist also for 3<=N<=5. The same calculi can be constructed using a method
introduced by Hermisson. In case N=3, the result is in accordance with the
known result by Apel and Schm\"udgen for the Podles sphere.
  In the second part, higher order differential calculus and symmetry are
treated. The relations which hold for the two-forms in the universal higher
order calculus extending one of the two first order calculi are given. A
"braiding" homomorphism is found. The existence of an upper bound for the order
of differential forms is discussed.
