Polynomial B-splines of given order m and with knots of arbitrary multiplicity are investigated with respect to their Chebyshev norm. We present a complete characterization of those B-splines with maximal and those with minimal norm, compute these norms explicitly, and study their behavior as m tends to infinity. Furthermore, the norm of the B-spline corresponding to the equidistant distribution of knots is studied. Moreover, we investigate the behavior of the B-spline's maximum, if a new knot is inserted and/or if one of the knots is moved. Finally, we analyse those types of knot distributions for which the norms of the corresponding B-splines converge to zero as m ¿ 8.