This paper looks at the topic of fractal geometry and chaos theory. It explores the basics of both fractal geometry and chaos theory. The paper also focuses on the Mandelbrot set and the Julia set. This paper is aimed at high school juniors and seniors. After reading this paper they should be able to begin to understand chaos theory and fractal geometry. The paper includes a review of The Julia Sets and Chaos: The Software, the two computer programs that are used in the accompanying extended lesson plan. This lesson plan is to help students more fully develop the ideas and understandings of the topics addressed in the paper. By the end of this project the students should have a good feeling of how mathematics is used today and that it is not a dead science.