  We consider the generalizations of the free U(N) and O(N) scalar conformal
field theories to actions with higher powers of the Laplacian box^k, in general
dimension d. We study the spectra, Verma modules, anomalies and OPE of these
theories. We argue that in certain d and k, the spectrum contains zero norm
operators which are both primary and descendant, as well as extension operators
which are neither primary nor descendant. In addition, we argue that in even
dimensions d <= 2k, there are well-defined operator algebras which are related
to the box^k theories and are novel in that they have a finite number of
single-trace states.
