Abstract: Julia sets are one of the most famous and beautiful examples of fractals, arising from a simple iterative rule applied to the complex numbers. I will discuss basic properties of Julia sets from the dynamical systems perspective. To explore some topological properties of Julia sets and the connection with the famous Mandelbrot set, I will prove the “fundamental theorem of the Mandelbrot set”. Time permitting, I will explain why the Mandelbrot set is connected and discuss a very recent and amusing result on Julia sets.