We will talk about some of the issues that arise in solving eigenvalue problems numerically, and introduce some recent ideas on how to address them. First we briefly demonstrate why we want to use methods different from how we first learn to solve eigenvalue problems by hand in linear algebra. Then we will talk about the power method, one of the most basic and powerful but sometimes very slow iterative methods. We will introduce momentum-type extrapolation methods which recycle older information in our approximation sequence to accelerate an iterative method. Finally, we will show some recent results on accelerating the power method using a dynamically chosen momentum term.