My goal for this talk is to introduce all of you to my research area and the work I did in my Ph.D. thesis. I'll start by talking about the beautiful and intricate Mandelbrot set, the object that inspired much of the modern field of complex dynamics. Then I'll give a brief introduction to the p-adic numbers. Finally, I'll put the two together to define a p-adic Mandelbrot set and spend a bit of time discussing what I've discovered about these objects. I won't assume any prior knowledge about the Mandelbrot set or about p-adic numbers, so feel free to encourage your students to attend.