One of the most famous (and still not fully understood) objects in mathematics is the Mandelbrot set.  By definition, it is the set of complex numbers c for which the recursive sequence defined by x_1 = c and x_{n+1} = (x_n)^2+c is bounded.  This set turns out to be rich and complicated and related to many different areas of mathematics.  In the first talk, I will present an overview of what's known and what's not known about the Mandelbrot set, and I'll describe recent work that (perhaps surprisingly) employs tools from number theory and ar