In this paper, we propose a theoretical study of the approximation properties of NURBS spaces, which are used in Isogeometric Analysis. We obtain error estimates that are explicit in terms of the mesh-size h, the degree p and the global regularity, measured by the parameter k. Our approach covers the approximation with global regularity from C0 up to Ck-1, with 2k - 1 ≤ p. Notice that the interesting case of higher regularity, up to k = p, is still open. However, our results give an indication of the role of the smoothness k in the approximation properties, and offer a first mathematical justification of the potential of Isogeometric Analysis based on globally smooth NURBS. © 2010 Springer-Verlag.