AbstractWe strengthen a theorem of Gitik and Shelah [6] by showing that if κ is either weakly inaccessible or the successor of a singular cardinal andSis a stationary subset of κ such thatNSκ↾Sis saturated then κ ∖Sis fat. Using this theorem we derive some results about the existence of fat stationary sets. We then strengthen some results due to Baumgartner and Taylor [2], showing in particular that ifIis aλ+++-saturated normal ideal onPκλthen the conditions of beingλ+-preserving, weakly presaturated, and presaturated are equivalent forI.