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There are currently 13 or 14 known artificial satellites in Mars orbit (6 active):

Mars-2, launched 1971 May 19, USSR, inactive

Mars-3, launched 1971 May 28, USSR, inactive

Mariner 9, launched 1971 May 30, USA, inactive

Mars-5, launched 1973 July 25, USSR, inactive

Viking 1, launched 1975 August 20, USA, inactive, estimated to deorbit 2019 or later

Viking 2, launched 1975 September 9, USA, inactive

Phobos-2, launched 1988 July 12, USSR, inactive

Mars Global Surveyor, launched 1996 November 7, USA, inactive

2001 Mars Odyssey, launched 2001 April 7, USA, active

Mars Express, launched 2003 June 2, European Space Agency (launched by Russia), active

Mars Reconnaissance Orbiter, launched 2005 August 12, USA, active

Mars Orbiter Mission (MOM) or Mangalyaan, launched 2013 November 5, India, active

Mars Atmosphere and Volatile Evolution (MAVEN), launched 2013 November 18, USA, active

ExoMars Trace Gas Orbiter, launched 2016 March 14, ESA and Russia, active

One small clarification: Mark Adler has pointed out that since we do not actively track the inactive orbiters, they may not actually be in orbit. They are generally expected to be in orbit longer, however.

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