Over the past twenty-five years, condensed matter physics has been developing materials with novel electronic characteristics for a wide range of future applications. Two research directions have shown particular promise: topological insulators, and high temperature copper based superconductors (cuprates). Topological insulators are a newly discovered class of materials that can be manipulated for spintronic or quantum computing devices. However there is a poor spectroscopic understanding of the current topological insulators and emerging topological insulator candidates. In cuprate superconductors, the challenge lies in raising the superconducting transition temperature to temperatures accessible in non-laboratory settings. This effort has been hampered by a poor understanding of the superconducting mechanism and its relationship with a mysterious pseudogap phase. In this thesis, I will describe experiments conducted on topological insulators and cuprate superconductors using scanning tunneling microscopy and spectroscopy, which provide nanoscale spectroscopic information in these materials.