Imagine you are a tourist, visiting New York for the first time, going for a stroll in Manhattan. Since you don't know the city, at each crossroad you decide at random where to go next - left, right or straight - with one restriction: you never want to go back to a place you've visited before. After n crossroads, how far will you be from your starting point? What kind of path will you have walked along? In this talk, we will present the probabilistic model for self-avoiding walk, and tell you what probabilists know about the answers to these questions.