The　directional-curvature　theory　is　developed　as　a　rational　basis　for　the　strain　energy　and　the　chemical　reactivity　in　single-walled　carbon　nanotubes　(SWCNTs)　and　fullerenes.　The　directional　curvature　KD　and　its　mean　KM,　derived　from　this　theory,　cover　the　overall　curvatures　of　their　bonds　and　atoms　and　break　through　the　limitations　of　the　pyramidalized-angle　θp　approach,　which　is　only　available　to　atomic　curvature.　The　directional-curvature　theory　demonstrates　that　KD　and　KM　depend　directly　on　the　strain　or　reactive　binding　energies　of　the　bonds　and　atoms　and　that　there　is　approximate　curvature　conservation　in　SWCNTs　and　fullerenes.　Application　of　this　theory　to　addition　reactions　of　various　SWCNTs　and　fullerenes　shows　that　the　slope　of　the　straight　line　between　the　strain　or　binding　energies　and　KD　is　close　to　a　constant,　which　helps　clarify　the　puzzle　as　to　why　some　functionalizations　of　C70　occur　at　the　relatively　flat　midsection.　©　2007　Wiley-VCH　Verlag　GmbH　&　Co.　KGaA.