  This paper studies eigenvalues of the buckling problem of arbitrary order on
compact domains in Euclidean spaces and spheres. We prove universal bounds for
the $k$-th eigenvalue in terms of the lower ones independent of the domains.
Our results strengthens the recent work by Jost, Li-Jost, Wang and Xia and
generalizes Cheng-Yang's recent estimates on the buckling eigenvalues of order
two to arbitrary order.
