The aim of this article is to discuss the generalized smoothness for the splines on q-covered manifold, where q is the natural number. By using mentioned smoothness it is possible to consider the different types of smoothness, for example, the integral smoothness, the weight smoothness, the derivatives smoothness, etc. We find the necessary and sufficient conditions for calculation of basic splines with a’priori prescribed smoothness. The mentioned smoothness may contain no more than q (locally formulated) linearly independentconditions. If the number of the conditions is exactly q, then the discussed spline spaces on the embedded grids are also embedded.