  An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined
which is covariant under U_q(so(2,D-1)), for q a root of unity. The
star-structure is studied in detail. The scalar fields have an intrinsic
high-energy cutoff, and arise most naturally as fields on orbifolds AdS_q^D
\times S^D/G if D is odd, and AdS_q^D \times S_{\chi}^{2D-1}/G if D is even.
Here G is a finite abelian group, and S_{\chi} is a certain ``chiral sector''
of the classical sphere. Hilbert spaces of square integrable functions are
discussed. Analogous results are found for the q-deformed sphere S_q^D.
