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We don't really know. What we do know is this:
The Torchwood Institute (Torchwood One), as it was prior to the Battle of Canary Wharf in "Army of Ghosts"/"Doomsday", is not. Torchwood Four has been missing for an unspecified time, so isn't active, either. Torchwood Three hasn't been active as an organisation since the "Children of Earth" incident, although two of its officers (Jack Harkness & Gwen Cooper) are still around. The current status of Torchwood Two, run by Archie (surname unknown), whom even Captain Jack thinks is "very strange", hasn't been revealed.
Torchwood has always been a secretive organisation, so it's impossible to be sure that there are no bits of it still fuctioning. Because nobody knows what happened to Torchwood Four, it's possible that it might turn up again -- Jack indicated as much to Gwen in "Everything Changes" (Torchwood story).
In Big Finish, Torchwood Three has reformed as late as 2017.