However, NASA's Cassini spacecraft orbited Saturn for 13 years (2004-2017) and provided an incredible wealth of data about the planet and its rings, moons, and atmosphere. Cassini is no longer operational, but it's legacy of data continues to be analyzed by scientists.
While no current spacecraft are specifically orbiting Saturn, future missions are being planned.
It's important to note that Saturn has many natural satellites (moons), some of which are in orbit around the planet. However, these are not artificial satellites like the ones we send from Earth.