The mystery of high temperature superconductivity is undoubtedly one of the great contemporary physics puzzles, the solution of which will open a vista of energy applications. We study the orbits traversed by electrons in the so-called 'normal' state of unconventional superconductors by subjecting them to magnetic fields as high as 10,000 times that of a bar magnet. I will discuss our findings from quantum oscillation measurements in a recently discovered 'iron pnictide' high temperature superconductor, and one of the original 'cuprate' high temperature superconductors, and consider implications for the mechanism of high temperature superconductivity.