The introduction of spectroscopic imaging scanning tunneling microscopy (SI-STM) has revolutionized our ability to image complex electronic quantum matter at atomic scale. I will review here the advances in visualization and understanding of the electronic structure of cuprate superconductivity. A comprehensive and consistent picture of this highly complex state of electronic matter emerges. It reveals a fundamentally bipartite electronic structure with heterogeneous quasi-localized high-energy states dominated by dopant-induced electronic disorder, and spatially homogeneous low energy momentum-space states which are the excitations of Cooper pairs. We explore the evolution of all these phenomena as the carrier density of hole-doped cuprates is reduced from the robust high temperature d-wave superconductor towards the non superconducting Mott insulator phase. We also use high precision Fourier-transform scanning tunneling spectroscopy (FTSTS) to visualize the momentum-space excitations in the mysterious ‘pseudogap’ phase of the cuprates. We report a continuous temperature dependence of the complete ‘octet’ of quasiparticle interference signals from T<0.1Tc to T>1.25Tc. A comprehensive and detailed understanding of all states with E<50meV and for a full reciprocal unit cell is achieved. Using these data, we can identify definitively the low temperature state of the cuprate ‘pseudogap’ regime which dominates transport and quantum oscillation phenomena. Finally we explore the distinct identity of the higher energy excitations which dominate the high temperature thermodynamic characteristics of these materials.