Yes, certainly, around 6 of 8 planets in the Solar System, the oldest of which, The 2001 Mars Oddessey orbiter has been in orbit around another planet longer than any other satellite.
A key thing to note is that artificial satellites are defined as anything man-made that orbits another celestial body, the difference with a space probe being that a satellite has to enter an orbit around a celestial body while a space probe does not. As such, the Apollo missions would've been temporary satellites of the Moon.
A list of extra-terrestrial orbiters can be found on Wikipedia
Below is a list of active (or prominent where none are active) orbiters on other planets from the above wikipedia list (As of Febuary 2018):
Mercury
- MESSENGER - the First Mercury orbiter was deliberately crashed in 2015 after 4 years in orbit. The orbiter achieved 100% mapping of Mercury by 2013. MESSENGER is credited with the characterisation of Mercury's magnetic field, and the discovery of water ice at the planets North Pole
Venus
- Akatsuki - the 8th orbiter around Venus entered orbit in late 2015. Its main mission is cloud and surface imaging from orbit around Venus to investigate is meteorology.
Mars
- 2001 Mars Odyssey - The longest extra-terrestrial orbiter, 2001 Mars Odyssey provided information about the existence of hydrogen and mapped the distribution of water below the shallow surface, amongst other experiments. It also served as the primary means of communications to various Mars surface explorers
- Mars Express - the first ESA orbiter for Mars, its primary mission is the orbital study of the interior, sub-surface, surface and atmosphere, and environment of the planet.
- Mars Reconnaissance Orbiter - Its primary goal being to map the Martian landscape with high-resolution cameras in order to choose landing sites for future surface missions, the orbiter also provided navigational data during EDL and acts as a communications relay for the Phoenix lander and MSL. The orbiter is also using its onboard equipment to study the Martian climate, weather, atmosphere and geology, and to search for signs of liquid water. The orbiter has also searched for the lost landers Beagle 2 and Mars Polar Lander, the former of which it found in 2015.
- MAVEN - another NASA orbiter, its four primary missions consist of determining the role that loss of volatiles to space from the Martian atmosphere has played through time, the current state of the upper atmosphere, ionosphere, and interactions with the solar wind, the current rates of escape of neutral gases and ions to space and the processes controlling them and the ratios of stable isotopes in the Martian atmosphere.
- Mars Orbiter Mission - The first orbiter and interplanetary mission for the ISRO, its main objectives are to develop the technologies required for designing planning and managing of interplanetary missions. The scientific objectives include the study of the morphology, topography and mineralogy of Mars' surface, studying the constituents of Mars' atmosphere and studying the dynamics of the upper atmosphere.
- ExoMars Trace Gas Orbiter - Initially tasked with providing telemetry for the Schiaparelli demonstration lander for 8 sols after landing, the orbiter then moved into a more circular orbit to carry out its scientific objectives which include; mapping Hydrogen levels to a depth of 1m below the surface, and the characterisation of spatial, temporal variation, and localisation sources of atmospheric trace gases
Jupiter
- Juno - Juno carries a host of science equipment to do various experiments, such as measure the abundance of water in Jupiter, obtain better estimates of its core mass, map its gravitational and magnetic fields, amongst other things
Saturn
- Cassini - The first and only orbiter of Saturn, which had a Grand Finale take place last year after 13 years in orbit. The orbiter was tasked with various missions, such as studying Saturn's rings, the composition of the surfaces of the natural satellites of Saturn, measuring the three dimensional structure and dynamic behaviour of the magnetosphere amongst other things.
Minor planets, asteroids and comets
- NEAR Shoemaker, 433 Eros - NEAR Shoemaker was tasked with studying the near-Earth asteroid 433 Eros, the data from the asteroid may have been used to establish a connection between Eros and other S-type asteroids and meteorites found on Earth.
- Dawn, 4 Vesta - Dawn estimated the size of Vesta's core as well as having observed gullies that are believed to have been formed by transiently flowing liquid water.
- Dawn, Ceres - At Ceres, Dawn captured a full topographic map of Ceres and continued further surveying of the dwarf-planet
- Rosetta, 67P/Churyumov-Gerasimenko - It's main objective was to map 67P to find a landing site for it's lander Philae.
