Eredeli's Bar (ერედელის ბარი), N. Baratashvili St. 1 (Down the stairs to a door with no sign and no name on the diagonal street just next to Europe Square (formerly Era Square) where the monument to Medea and the Golden Fleece is.). A very cool bar that attracts the cool locals and expat English teachers. There are no signs and you might have to ring the doorbell on nights with live music, but it's not a private club. Very nice and friendly staff and they have the best cha cha I've tasted. The Caesar salad is also excellent. It's also a gallery and it has WIFI.
