Abstract: Topological quantum matter (TQM), where topological order or topological invariants are used to distinguish different phases of matter, has emerged as a major paradigm in condensed matter physics in recent years. TQMs feature topological bulk-boundary correspondences, where some nontrivial topologically-protected boundary modes are guaranteed to emerge due to the topologically nontrivial states in the bulk of the system. The first example of TQM is the well-known quantum Hall (QH) effect of two-dimensional electrons in a perpendicular magnetic field, where the bulk is insulating due to energy gaps from Landau level formation, and topological conduction free of backscattering occurs via chiral edge states, giving rise to quantized Hall conductance in units of e2/h that is now used as a quantum metrology to help define “ohm” or the even Planck constant itself. The list of TQMs has dramatically expanded in the past decade to now include new states of matter such as topological insulators (TI), which can be a generalization of the QH states to three dimensions and zero magnetic field due to the presence of strong spin orbit coupling (SOC), giving rise to a gapped insulator in the bulk and conducting spin-helical Dirac fermions on the surface promising for spintronics and other applications; topological semimetals, which realize 3D Dirac or Weyl fermions that can exhibit a condensed matter version of the “chiral anomaly”; topological superconductors, which could host quasiparticle analogues of “majorana fermions” potentially useful as qubits for “topological” quantum computation. While so far mostly studied for electronic systems, it is also possible to engineer “synthetic” gauge fields or SOC that may help realize analogous or new kinds of TQMs for photons or neutral atoms. This talk will overview some of the key physics and promised device applications, and describe efforts in my group to make, improve and characterize TQMs --- a particular focus in the past few years has been to realize truly intrinsic TIs that demonstrate salient signatures of “topological” transport, such as a thickness independent conductance in thin films, “half-integer” Dirac fermion QH effects and helical spin polarized current characteristic of topological surface states (TSS), and a “half-integer” Aharonov-Bohm effect when such TSS are confined in a (cylindrical) curved space. Such TIs could also be used as a starting point to make topological semimetals and superconductors.