Expanding your question to 2008 (as it is now as I'm answering, so up until the last episode of Season 4 of the new series, Journey's End)... There have been 202 stories (if you include the unfinished Shada) comprising 757 episodes. And if you don't count Shada, 201 stories 751 episodes.
Including Series 7 (which airs later this year), the 11th Doctor has been in 45 episodes of Doctor Who and is played by Matt Smith.
Doctor Who is a British science fiction television programme produced by the BBC. The programme depicts the adventures of a mysterious alien time-traveller known as "the Doctor" who travels in his space and time-ship, the TARDIS, an acronym for Time And Relative Dimension(s) In Space. The TARDIS is much larger on the inside than on the outside and due to a malfunction of its Chameleon Circuit, is stuck in the shape of an old 1950s-style British police box... With his companions, he explores time and space, solving problems and righting wrongs along the way.Doctor Who is a Time Lord from the planet Gallifrey.The programme is listed in Guinness World Records as the longest-running science fiction television show in the world.The show's lead character is currently portrayed by David Tennant (2005 - 2008)Doctor Who first appeared on BBC television at 5:15 pm(GMT) on 23 November 1963.And has been played by the following people So Far...First Doctor, played by William Hartnell (1963-1966) (136 episodes)Second Doctor, played by Patrick Troughton (1966-1969) (127 episodes)Third Doctor, played by Jon Pertwee (1970-1974) (129 episodes)Fourth Doctor, played by Tom Baker (1974-1981) (173 episodes)Fifth Doctor, played by Peter Davison (1981-1984) (70 episodes)Sixth Doctor, played by Colin Baker (1984-1986) (35 episodes)Seventh Doctor, played by Sylvester McCoy (1987-1989, 1996) (42 episodes)Eighth Doctor, played by Paul McGann (1996) (35 episodes)Ninth Doctor, played by Christopher Eccleston (2005) (13 episodes)Tenth Doctor, played by David Tennant (2005-2009)Eleventh Doctor, played by Matt Smith (2010-2013)Twelfth Doctor, played by Peter Capaldi (2013-present)Other People have played the doctor, but these were mainly in feature length movies animations, stage plays, audio recordings and not forgetting spoof versions of the character as well.The Doctor over the years has had approx 35 companions helping him with his adventures.
The first episode of Doctor Who to be broadcast in colour was the first part of the Third Doctor's first story, 'Spearhead from Space', broadcast on the January 3rd 1970. Since then all episodes of Doctor Who have been broadcast in colour, with the 50th anniversary special, 'The Day of the Doctor', being the first adventure to be broadcast in 3D on November 23rd 2013.
There has been 13 actors who has played the role of the Doctor on Doctor Who. Also Peter Capaldi, who is currently playing the Doctor, is also the new 1st Doctor of the new 12 count regeneration cycle.
Since 1963 to date
There have been 753 episodes in Doctor Who.
there has been about 777 episodes of Doctor Who
There are 696 episodes of classic Doctor Who (not including the TV movie), comprising 157 different stories. However a large number of these episodes (108 in total) are 'lost' and it is thought copies no longer exist. The First Doctor: 29 stories (134 episodes) The Second Doctor: 21 stories (119 episodes) The Third Doctor: 24 stories (128 episodes) The Fourth Doctor: 42 stories (172 episodes) The Fifth Doctor: 20 stories (69 episodes) The Sixth Doctor: 8 stories (31 episodes) The Seventh Doctor: 13 stories (43 episodes)
Doctor Who began airing in 1963, which seasons of between 14 and 45 episodes each (depending on the year). It stopped in 1989. A television movie was made in 1996. Doctor Who was revived for a modern audience in 2005, and has been going since.
796
The Doctor Who episodes have been written by different writers.
Ken Tyllssen has: Played Dalek in "Doctor Who" in 1963. Played 1st Sensorite in "Doctor Who" in 1963. Played First Sensorite in "Doctor Who" in 1963. Played Mechonoid in "Doctor Who" in 1963. Played First Scientist in "Doctor Who" in 1963. Played Mechanoid in "Doctor Who" in 1963. Played Assistant in "Doctor in the House" in 1969.
Adrian Gibbs has: Played Rysik in "Doctor Who" in 1963. Played Extra in "Doctor Who" in 1963. Played The Watcher in "Doctor Who" in 1963. Played Soldier in "A Bridge Too Far" in 1977. Performed in "A Man Called Intrepid" in 1979. Played Himself - The Watcher in "A New Body at Last" in 2007.
Including Series 7 (which airs later this year), the 11th Doctor has been in 45 episodes of Doctor Who and is played by Matt Smith.
Andrew Staines has: Played Sergeant to Benik in "Doctor Who" in 1963. Played Goodge in "Doctor Who" in 1963. Played Keaver in "Doctor Who" in 1963. Played Captain in "Doctor Who" in 1963. Played Clerk of Magistrates in "Justice" in 1971. Played Mr.Wansborough in "The Woman in White" in 1982. Played Goodge in "Rogue Time Lords" in 2007. Played Prof Goodge in "The UNIT Family: Part Two" in 2011.
Murray Grumbar has: Played Arcturus in "Doctor Who" in 1963. Played Dalek in "Doctor Who" in 1963. Played Dalek Machine Operator in "Doctor Who" in 1963. Played Dalek Operator in "Doctor Who" in 1963. Played Mechonoid in "Doctor Who" in 1963. Played Mechanoid in "Doctor Who" in 1963. Played Dining Car Attendant in "Moody and Pegg" in 1974.
Barney Lawrence has: Played Android in "Doctor Who" in 1963. Played Royal Servant in "Doctor Who" in 1963. Played Guard in "Doctor Who" in 1963. Played Foster in "Doctor Who" in 1963. Played Seabase Guard in "Doctor Who" in 1963. Played Marshman in "Doctor Who" in 1963. Played Sailor on the Shadow in "Doctor Who" in 1963. Played 2nd Kinda Hostage in "Doctor Who" in 1963. Played Dave Culshaw in "Doctor Who" in 1963. Played Guard in "Blakes 7" in 1978. Played Federation Trooper in "Blakes 7" in 1978.