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.