Although the kebabs above are known throughout China, the dish which most people rightly associate with Xi'an is this, even if they have never actually eaten it.
Image by NNU-1-05100104 from Wikipedia. This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license*.
羊肉泡馍 (yáng ròu pào mó), literally 'sheep meat soaked bread' is a stew containing lamb or mutton in a lamb broth with torn up bread (馍 - mó) and noodles. A hearty, filling dish which is very welcome in Xi'an's bitter winter. Usually served with pickled or raw garlic and chilli sauce.
Recipe here.
There is an alternative, less common version which uses beef instead - 牛肉泡馍 (niúròu pàomó). The best place to try this is in the Muslim Quarter.
* To my surprise, I don't seem to have a picture of the dish, despite having eaten it literally hundreds of times. I would have gone out to the local Xi'an restaurant and had some for lunch, but we are in our 5th day of appalling non-stop monsoon rains. I'll replace the Wikipedia picture with one of my own later.