Let X, Y be nonsingular real algebraic sets. A map φ : X → Y is said to be k- regulous, where k is a nonnegative integer, if it is of class Ck and the restriction of φ to some Zariski open dense subset of X is a regular map. Assuming that Y is uniformly rational, and k ≥ 1, we prove that a C∞ map f : X → Y can be approximated by k-regulous maps in the Ck topology if and only if f is homotopic to a k-regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking Y = Sp (the unit p-dimensional sphere), we obtain several new results on approximation of C∞ maps from X into Sp by k-regulous maps in the Ck topology, for k ≥ 0