Nanometer-scale electronic transport in engineered interfaces in ferroelectrics, such as domains and topological defects, has emerged as a topic of broad interest due to potential applications in information storage, sensors and photovoltaic devices. Scanning probe microscopy (SPM) methods led to rapid growth in the field by enabling correlation of the unique functional properties with microstructural features in the aforementioned highly localized phenomena. In addition to conduction localized at interfaces, polarization-mediated control of conduction through domains in nanoscale ferroelectrics suggests significant potential for use in memristor technologies. In parallel with experiment, theory based on thermodynamic Landau-Ginzburg-Devonshire (LGD) framework has seen rapid development, both rationalizing the observations, and hinting at possibilities for local, deterministic control of order parameters. These theories can successfully account for static interface conductivity at charged, nominally uncharged and topologically protected domain walls. Here, recent experimental and theoretical progress in SPM-motivated studies on domain wall conduction in both standard and improper ferroelectrics are reviewed. SPM studies on transport through ferroelectrics reveal that both domains and topological defects in oxides can be exploited as individual elements for use in functional nanoscale devices. Future prospects of the field are discussed.