Trastevere is one of the most romantic neighborhoods in Rome, located on the west bank of the Tiber river. There are so many picture-like and hidden alleys in this neighborhood, leading to beautiful piazzas. This is one of the oldest areas of Rome and these days there are quite foreign academic institutes and therefore more expats and students.
There are great restaurants in Trastevere, some might be too touristy.
Better search for good and authentic restaurants, such as Via Delle Frattei, where the owner himself can sometimes serve you lunch