This chapter presents some of the mathematics behind one of the most intricate and interesting shapes in dynamics, the Mandelbrot set. The filled Julia sets seemed to fall into one of two classes: those that were connected and those that were totally disconnected. When parameters are varied, real functions occasionally encounter bifurcations, most often saddle-node or period-doubling bifurcations. The chapter looks at these bifurcations in the complex plane. From a global perspective, there is also a major change in the structure of the Julia sets as the parameter passes through a period-doubling bifurcation.