Abstract: Let G = G’ be either a simple algebraic group over an algebraically closed field k or a finite Chevalley group (normal or twisted) over k = F, with q > 4. For the case of an algebraic group all groups discussed are taken to be closed subgroups of G. In this paper we prove results that parallel those in [ 141, where we deterined those subgroups of G containing a maximal torus (given restrictions on k). Here we study the overgroups of a certain type of unipotent subgroup of G which like the maximal tori are of particular importance to the understanding of the subgroup structure of G. As applications we establish results about overgroups of local subgroups of G and results concerning generation by centralizers of unipotent elements. A unipotent group I’< G is said to be full provided I’ contains each unipotent element of its centralizer. Such subgroups occur quite often. For example, let V,, be an arbitrary unipotent subgroup of G and V a maximal unipotent subgroup of either NG( V,) or V,C,( V,). Then V, < V and V is full. Before stating the results, we introduce the following notation and terminology. Let p = char(k). A subgroup X of G is quasisimple if it is a perfect central extension of a simple group (connected in case G is an algebraic group); X is semisimple if it is a commuting product of quasisimple groups. Let R,(X) denote the unipotent radical of X, which we interpret as O,(X) in case G is finite. If k = iF, or F4, let O”‘(X) denote the group generated by all unipotent elements (i.e., p-elements) of X.