  Recent advances in creating complex oxide heterostructures, interfaces formed
between two different transition metal oxides, have heralded a new era of
materials and physics research, enabling a uniquely diverse set of coexisting
physical properties to be combined with an ever- increasing degree of
experimental control. Already, these systems have exhibited such varied
phenomena as superconductivity, magnetism, and ferroelasticity, all of which
are gate-tunable, demonstrating their promise for fundamental discovery and
technological innovation alike. To fully exploit this richness, it is necessary
to understand and control the physics on the smallest scales, making the use of
nanoscale probes essential. Using the prototypical LaAlO$_3$/SrTiO$_3$
interface as a guide, we explore the exciting developments in the physics of
oxide-based heterostructures, with a focus on nanostructures and the nanoscale
probes employed to unravel their complex behavior.
