On a closed (i.e. compact, no boundary) odd-dimensional manifold, the index of any elliptic differential operator is zero. Thus, on odd-dimensional manifolds, interesting examples can only be obtained by dropping either "elliptic" or "differential". This talk will explain the two cases. If "differential" is dropped, then the examples are Toeplitz operators (which are elliptic pseudo-differential operators of order zero). A corollary is the result of Louis Boutet de Monvel about Toeplitz operators associated to the boundary of a strictly pseudo-convex domain. If "elliptic" is dropped, then examples are the differential operators on contact manifolds studied by E. van Erp, Baum-van Erp, and Gorokhovsky-van Erp.