Information
The Republic Of Luna, circa 3358 A.D.
At this point, not much is known about the 34th Century. However, some information has been revealed, such as the Moon is now an independent nation called the Republic Of Luna. It also seems that delegations from other star systems, such as Vega, are common (it is not clear if these delegations are from a human colony world, or an alien one).
Time travel
By the 34th Century, time travel has become common. Many historians routinely travel into the past to study it. At times, they even interact with the people they meet there, although they are not permitted to reveal too much about the future, for fear of damaging the time line.
One such historian is Jenkalu Hannah Buckland (or Kalu as she is also known as), a descendant of Rex Buckland and Hannah Webster.
On several occasions, Kalu has had the privilege of meeting her famous ancestors and even helped them on several occasions.
The Temporal Enforcers
All time travel is regulated by the Temporal Enforcers, a police force charged with monitoring the time stream. When an attempted change is detected, the Temporal Enforcers have a free hand to act to prevent the change from taking place. This includes recruiting people from the past, although this usually only happens when the people in question will be themselves affected. This happened to Rex and Hannah in 2009.
Good and evil at peace
Another aspect of the 34th Century is that magic is no longer a secret and that Good and Evil now peacefully co-exist side by side. This was accomplished in no small part due to the events which started with Rex and Hannah leading the Warlock Insurgency from 2001 to 2006.
In 2009, Rex and Hannah inspired two Demons named Kalton and Zova to encourage change in the Underworld. The 34th Century peace between Good and Evil are the long term results of the events that Rex, Hannah, Kalton, and Zova had set in motion, more than a thousand years earlier.
Also responsible were the descendants of the Halliwell sisters and Charmed Sons.
When the Demon, Zankou, escaped from the Demonic Wasteland, after being imprisoned there for more than a thousand years, he was appalled at this lasting peace between both sides. He saw it as an abomination. When he tried to stir up trouble in the Underworld, no one took him seriously, rather he was regarded as a relic of a bygone era. He thus made an ill fated attempt to alter the past, but was thwarted.
Future
It remains to be seen how much more of the 34th Century will be revealed in the future.
References
Rex and Hannah Chronicles Story No. 9: The Visitor
Rex and Hannah Chronicles Story No. 15: The Full Circle
Rex and Hannah Chronicles Story No.18: A Christmas Caper
Rex and Hannah Chronicles Story No. 30: Future Imperfect