Wikipedia gives as good a summary as any. Most of the breakdowns are minor, but (as of July 2022) scientists have had to use the backup computer since 2018.
Ongoing: Short circuitong in the drill's percussive mechanism
Since early 2015, the percussive mechanism in the drill that helps chisels into rock has had an intermittent electrical short circuit.
Resolved: Drill motor malfunction affecting the robotic arm
In December 2016, the motor inside the drill caused a malfunction that prevented the rover from moving its robotic arm and driving to another location. The fault is in the drill feed motor - internal debris is suspected. The fault was determined to be limited to the drill mechanism and the rover started moving again on December 9. The robotic arm is functional, and the Curiosity team performed diagnostics on the drill mechanism throughout 2017. On June 4, 2018, NASA announced that Curiosity's ability to drill has been sufficiently restored by changing the drilling methods.
Ongoing, but with a workaround: Cpmputer data storage issues
Since September 15, 2018, a glitch in Curiosity's active computer (Side-B) has prevented Curiosity from storing science and key engineering data. On October 3, 2018, the JPL began operating Curiosity on its backup computer (Side-A). Curiosity will store science and engineering data normally using its Side-A computer until the cause of the glitch in Side-B is determined and remedied.
Internal debris may be causing problem with Mars rover's drill. Dec 2016.
"NASA Is Trying to Get Mars Rover Curiosity's Arm Unstuck". Popular Mechanics. Associated Press. 13 December 2016. Retrieved 18 January 2017.
"Curiosity Mission Updates - Mars Science Laboratory". [Note: the link is constantly updated and no retrieval date was given by Wikipedia.]
Good, Andrew (4 June 2018). "Mars Curiosity's Labs Are Back in Action". NASA. Retrieved 4 June 2018.
Greicius, Tony (2 October 2018). "Curiosity Rover to Temporarily Switch 'Brains'". NASA. Retrieved 9 October 2018.