If you were planning to save a little money buying your PS5 by not picking up a pair or even a handful of controllers and simply re-using your existing PS4 ones, you're out of luck. The PS4 DualShock design will not work with most PS5 games and will only be useful for playing compatible PS4 games on the new-gen console.
Sony made an announcement about what controllers and accessories will be able to move over to the PS5 and it is quite extensive, covering everything that would be considered specialist. That includes licensed racing wheels, arcade sticks, flight sticks, and Platinum and Gold wireless headsets — all will work just fine on PS5 games. Third party headsets that connect via USB or 3.5mm audio jacks will also work on the PS5, but not necessarily wireless ones — the PlayStation headset companion app won't work with the PS5.
PS Move Motion controllers and the PSVR Aim controller will also work with supporting games on the PS5. So will the PlayStation Camera, though it will likely need a PlayStation Camera Adapter which will be given out for free to PS VR owners.
The DualShock 4, however, is out of luck when it comes to PS5 games.
"We believe that PS5 games should take advantage of the new capabilities and features we’re bringing to the platform, including the features of DualSense wireless controller," Sony said of this announcement, suggesting that new features of the PS5 gaming experience will need a new controller to enjoy in its entirety.