Abstract: Just what is the probability that a random walker has visited m sites after n steps? This seemingly innocent question has been the subject of intense research for over 70 years, going back to Erdős. We will review some(!) of the results in the area and explain the basic ideas behind the proofs. This will include a result of J. Liu and Q. Vogel for the planar case, published last year. We will also draw the audience's attention to a number of open problems in the area.