For some reason, Washington has several very good Middle Eastern Restaurants.
After having spent a year in Israel, I definitely start jonesing for a good falafel or shwarma from time to time. There are three places that come to mind immediately (although there are many others in DC).
The First place is called The Lebanese Taverna, and is a popular Lebanese Restaurant located on Connecticut Avenue, directly across the street from the Woodley Park Metro Stop. This place gets crowded, but is worth a wait.
The Second Restaurant is open 24 hours and is quite a dive. It is called the DC Cafe, and is located on P street, two blocks from DuPont Circle. I would go here to practice my Arabic, and get good falafel. After a night out on the town, this is a great place to end your evening.
The last place is not exactly in DC, but is my favorite. This is an Israeli owned kosher restaurant in Silver Spring, MD, and takes about 20 minutes to get to. They have the BEST shwarma and falafel in the area. They also have a very Israeli attitude, which is always a bit amusing (and a bit obnoxious). Max's is located on University Avenue (which splits off of Connecticut Ave once it hits MD). While this may be a bit out of your way, if you have the time, or a serious desire for Israeli fast food...go to Max's.
Close