The author investigates the problem of the existence of monotone and/or convex splines which have degree n and order of continuity k and which interpolate a given set of data. Where they exist, the author gives a method for obtaining the desired splines by using Bernstein polynomials of appropriate piecewise linear functions. A number of algorithms involving the given data are developed which indicate the existence or non-existence of a solution to the problem with appropriate monotone or convex characteristics, or both. In all cases considered, a necessary condition for a solution to exist is that

$k\le n-k$. These results generalize results due to

*D. F. McAllister, E. Passow* and

*J. A. Roulier* [Math. Comput. 31, 717-725 (1977;

Zbl 0371.65001)] and

*E. Passow* and

*J. A. Roulier* [SIAM J. Numer. Anal. 14, 904-909 (1977;

Zbl 0378.41002)].