The L^p spaces are well-studied in classical harmonic analysis, where p is always a positive constant. When we allow p to be a function as well, we then have variable L^p spaces. In the last few years, my three main projects have been on the variable analogues of Hardy spaces, extrapolation theory, and Muckenhoupt weights. In this talk, I will give a general overview of these three topics, starting with their classical theory and variable extensions, and summarizing the work I have done.