In the last 40 or so years, there has been a wealth of literature in both probability and combinatorics on random walks with small steps in the positive quadrant. In this setting one considers walks on the square lattice in which each step is to a horizontally, vertically or diagonally adjacent vertex, and each possible step is chosen with a fixed probability. I will focus on the enumeration problem, which is concerned with the probability that such a random walk of a given length stays entirely within the positive quadrant. For certain choices of the probabilities, the problem is exactly solved. I will discuss a method used by Bernardi, Bousquet-Mélou and Raschel involving elliptic functions, which applies to every solved case. I will also briefly describe how I have extended this method to enumerate four classes of walks by winding angle. I will not assume any familiarity with these problems on the part of the audience.
