I know Damascus has been around for at least 3500 years but I was wondering if there are any that are even older. By "still exists today" it doesn't necessarily have to be a huge city like Athens or Rome, but at least a functioning city that is known for more than just being an archaeological site.
Byblos in Lebanon is often cited as the oldest continually habituated city in the world.
Another city often given this title is Amesbury in Britain which is also noted for Stonehenge.
The earliest settlement in Damascus is nearly 9000 years old, not 3500, so it's pretty old. The earliest cities were located in the fertile crescent; due to the constant high population levels in these areas, a number remain inhabited. According to legend, Memphis in southern Egypt is nearly 5000 years old, whilst Jerusalem is probably not much older. There are a load of Mesopotamian cities which are very, very old, although few of these are still inhabited.