Suppose that G is a group of permutations of a set ${Omega}$. For a finite subset ${gamma}$of${Omega}$, the movement of ${gamma}$ under the action of G is defined as move(${gamma}$):=$maxlimits_{g{epsilon}G}|{Gamma}^{g}{ackslash}{Gamma}|$, and ${gamma}$ will be said to have restricted movement if move(${gamma}$)<|${gamma}$|. Moreover if, for an infinite subset ${gamma}$of${Omega}$, the sets|{Gamma}^{g}{ackslash}{Gamma}| are finite and bounded as g runs over all elements of G, then we may define move(${gamma}$)in the same way as for finite subsets. If move(${gamma}$)${leq}$m for all ${gamma}$${subseteq}$${Omega}$, then G is said to have bounded movement and the movement of G move(G) is defined as the maximum of move(${gamma}$) over all subsets ${gamma}$ of ${Omega}$. Having bounded movement is a very strong restriction on a group, but it is natural to ask just which permutation groups have bounded movement m. If move(G)=m then clearly we may assume that G has no fixed points is${Omega}$, and with this assumption it was shown in [4, Theorem 1]that the number t of G=orbits is at most 2m-1, each G-orbit has length at most 3m, and moreover|${Omega}$|${leq}$3m+t-1${leq}$5m-2. Moreover it has recently been shown by P. S. Kim, J. R. Cho and C. E. Praeger in [1] that essentially the only examples with as many as 2m-1 orbits are elementary abelian 2-groups, and by A. Gardiner, A. Mann and C. E. Praeger in [2,3]that essentially the only transitive examples in a set of maximal size, namely 3m, are groups of exponent 3. (The only exceptions to these general statements occur for small values of m and are known explicitly.) Motivated by these results, we would decide what role if any is played by primes other that 2 and 3 for describing the structure of groups of bounded movement.