  The search for topological phases of matter is evolving towards strongly
interacting systems, including magnets and superconductors, where exotic
effects emerge from the quantum-level interplay between geometry, correlation
and topology. Over the past decade or so, scanning tunnelling microscopy has
become a powerful tool to probe and discover emergent topological matter,
because of its unprecedented spatial resolution, high-precision electronic
detection and magnetic tunability. Scanning tunnelling microscopy can be used
to probe various topological phenomena, as well as complement results from
other techniques. We discuss some of these proof-of-principle methodologies
applied to probe topology, with particular attention to studies performed under
a tunable vector magnetic field, which is a relatively new direction of recent
focus. We then project the future possibilities for atomic-resolution
tunnelling methods in providing new insights into topological matter.
