\indent Discrete problems have a habit of being beautiful but difficult. In this talk we'll survey a menagerie of discrete analogues of classical operators arising in harmonic analysis, including singular integral operators, maximal functions, fractional integral operators, and Radon transforms. In general, discrete Radon transforms are the most difficult to treat, and we'll outline some of the methods that are breaking new ground in this direction. Key aspects of the methods presented come from number theory, and we will highlight the roles played by theta functions, exponential sums, Waring's problem, and the circle method of Hardy and Littlewood.