This　paper　devotes　to　the　adaptive　globally　synchronization　within　predefined-time　of　two　time-delayed　fractional-order　chaotic　systems.　Firstly,　through　fractional　calculus,　two　novel　different　fractional-order　systems　with　time-delay　are　proposed,　whose　convergence　is　guaranteed　and　phase　trajectory　is　given.　Secondly,　by　exploiting　the　non-negative　Lyapunov　function　and　inequality　theorem,　a　novel　global　predefined-time　stability　theorem　is　proposed,　which　can　ensure　the　settling　time　tunable.　And　the　upper　bound　of　the　settling　time　estimation　is　more　accurate　compared　with　the　classical　results.　With　the　help　of　novel　predefined-time　stability　theorem,　two　active　controllers　are　designed,　namely　the　fixed-time　synchronization　controller　and　predefined-time　synchronization　controller,　to　achieve　the　fixed-time　synchronization　and　the　predefined-time　synchronization　of　two　different　time-delayed　fractional-order　chaotic　systems　respectively.　Finally,　several　numerical　simulations　are　presented　in　order　to　show　the　effectiveness　of　the　proposed　methods.　©　2021　Institute　of　Electrical　and　Electronics　Engineers　Inc..　All　rights　reserved.