Yes, cats groom themselves, but they could use some help from their human friends to stay in tip-top shape.
We see cats licking their fur to stay clean. They do a pretty good job of it, but you can also help them.
Cats don’t really need a bath. If they do, use room temperature water, and place an oven rack in your sink or bathtub. The cat will cling to that instead of to your arm.
Brush your cat regularly. They’ll probably love the feeling of being groomed with a medium-bristle brush. Don’t brush against the grain of their hair, though. That will probably ruffle some proverbial feathers.
Regular brushing keeps their skin healthy, prevents matting, and reduces shedding and hairballs.
You also need to trim their nails, no matter how many scratching posts (or table legs) you have available for them.
If your cat has fleas, it might be a good time to see your vet or professional groomer to take care of the issue.