In this paper cyclic $n$-groupoids (Definition l), i.e., cyclic $n$-quasigroups (because every cyclic $n$-groupold is necessarily a cyclic $n$-quasi-group) are considered. Different equivalent definitions of a cyclic $n$-groupoid are given. Examples of cyclic $n$-quastgroupe are listed. Circular parastrophes of an $n$-quasigroup,which are suitable for the study of cyclic $n$-quasigroups, are defined. It is determined which parastrophes of a cyclic $n$-quasigroup are cyclic. It is shown that an $n$-quasigroup which is isotopic to a cyclic $n$-quasigroup must be isotopic to all its circular parastrophes and conditions under which its parastrophes are isotopic to a cyclic $n$-quasigroup are given. Some consequences which follow from the assumption that an $n$-quasigroup is isotopic to one of its parastrophes are obtained. Using these consequences a theorem which gives necessary and sufficient conditions for an $n$-quasigroup to be isotopic to a cyclic $n$-quesigroup is proved.