Modern quantum materials support a wide variety of exotic and unanticipated states of quantum matter and differ radically in phenomenology from conventional systems such as metals, semiconductors, band insulators, and ferromagnets. For example, quantum materials exhibit states such as electron liquid crystals, fluids of fractionalized quantum particles, quantum-entangled spin liquids, and topologically protected composite quantum particles. However, predictive theory is not fully developed for these forms of electronic quantum matter (EQM) and exploratory empirical research is required to discover and understand their properties. One of the most powerful and productive new techniques to achieve this is direct visualization of EQM at the atomic scale. For EQM, as with many highly complex systems in nature, seeing is believing and understanding. Here we describe the experimental, theoretical and analysis techniques of atomic-resolution spectroscopic imaging scanning tunneling microscopy (SI-STM) that allow such complex and enigmatic electronic/magnetic states to be directly visualized, identified, and understood. © Springer Nature Switzerland AG 2019.