Is it advisable to prime walls before painting even though they have a previously applied covering of paint?

No, just wipe them down to clean them. The primary function of primer is to seal the drywall or wood so it won’t absorb the more expensive paint. Priming an already painted surface only helps if the original paint is very dark, and you’re painting it a much lighter color, as it starts to lighten it up.

Depends on how dark they currently are, and how light color the paint is that you want to put on it.

Let’s say that your walls are dark blue now and you want to paint an off-white or vanilla color. You definitely want to prime your walls with an white coat.

On the other hand, if you are simply going white over white, it isn’t necessary.

Only if there will be a major change of color, like from light to dark or dark to light.
Watch for areas that might have been repaired and not adequately primed/painted.
It is advisable to thoroughly wash the walls before applying paint.

Also, if the existing paint is 10 years old, or older, use primer.

I had my house painted in TN 9 or 10 years after the first time it was painted. When painters painted my walls they had a tool like a long broom, but on the end was a flat thing that had sandpaper on it. They ran the sandpaper on the walls, like how you roll on paint, but it was just a quick thing they did, they didn’t worry about it being perfect end to end. Then they painted.

^^ Yes, roughing the surface of the old paint helps the new coat adhere better, so you won’t get big, gloppy runs.
The sanding only needs to be a slight roughing of the surface.
Primer is not necessary if you are not doing a complete removal of the old paint.

Some neighbors when I was growing up made a point of sanding everything thoroughly when re-painting because otherwise, “the room will get smaller every time you paint!”

Keep in mind that you may have to use 2 coats of paint, so no need to prime again, especially over contrasting colors.
Another thing: primer can be tinted to match color of paint to be applied!

It is advisable because it gives a surface that binds better than just applying to old psint

If you prime first you’ll find your coverage is better.