51st century

Ongoing

Another Ice Age affects Earth during this time. (DW: The Talons of Weng-Chiang, PDA: Emotional Chemistry).
Earth begins to be abandoned by the rich and powerful as the sun goes nova.
Those left behind have fractured into millions of nation-states ("Alliances") battling through the ether in the Transmat Wars. (DWM: The Keep).
At around this time, humanity first begins to develop time travel. The Supreme Alliance, at the initiative of its minister of justice Magnus Greel, experiments with time travel using a method based on Double nexus particles and zygma beam, but this proves to be a dead end. Magnus Greel was the first person to travel trough time, but became horribly disfigured for it. (DW: The Talons of Weng-Chiang)
The Time Agency operates out of this century, but it is closed again in same century, and only two agents remain. (DW: The Talons of Weng-Chiang, The Empty Child, TW: Kiss Kiss, Bang Bang)
The Doctor called this era a scientific dark age.

Major Events

Early 51st century - The Icelandic Alliance fell and the Supreme Alliance, of whom Magnus Greel was Minister of Justice, rose in its stead. The Supreme Alliance committed a number of war crimes and was brought down by the Filipino Army at the Battle of Reykjavik, which the Doctor witnessed. (DW: The Talons of Weng-Chiang)
Early 51st century - Magnus Greel kills approximately 100,000--a deed that earns him the epithet "Butcher of Brisbane".
unknown - The Time Agency formed and began employing Time Agents for missions in other time periods. (DW: The Empty Child) Later it was closed down and only seven agents were left. (TW: Kiss Kiss, Bang Bang)

Minor Events

5094 - Jack Harkness is voted "rear of the year" (TW: Kiss Kiss, Bang Bang)
unknown : The weapons factories of Villengard are destroyed by the Doctor and replaced by a banana grove. (DW:The Doctor Dances)
unknown - The spaceship SS Madame De Pompadour becomes disabled. Adrift in the Dagmar Cluster, its computer core begins cannibalizing the crew for parts; when this proves ineffective, it creates clockwork droids and attempts to kidnap the original Madame de Pompadour from the 18th century. (DW: The Girl in the Fireplace)
Unknown - The Doctor and Donna Noble visit The Library where they face an infestation by the Vashta Nerada, and the Doctor meets River Song for the first time in his personal timeline (but not in hers). The nature of The Library is revealed as the Doctor retrieves the consciousnesses of some 4,022 humans who were uploaded to the memory banks of the supercomputer, CAL. (DW: Silence in the Library / Forest of the Dead)

Notable individuals

Jack Harkness, John Hart, Gray and Magnus Greel all came from this century.

River Song is working in this century, and ultimately ends up preserved in the CAL computer during this time, but given her travels with the Doctor it's uncertain if she was actually a native of the century