Electronic states at boundaries of crystals, such as surfaces, interfaces, edges, hinges, corners, and extremities, play crucial roles in emerging quantum materials like graphene and similar monatomic-layer materials, van der Waals crystals, and topological insulators. Electronic states at such boundaries are different from those inside the three- or two-dimensional crystals, not only due to the truncation of crystal lattices, but also due to space-inversion-symmetry breaking and difference of topology in band structures across the boundaries. Such quantum materials are expected to advance energy-saving/-harvesting technology as well as quantum computing/information technology because of exotic phenomena such as electron’s spin-momentum locking, pure spin current, dissipation-less charge current, non-reciprocal current, possible Majorana fermions, and so on. In this review, their fundamental concepts are introduced from the viewpoint of surface physics, in which atomic and electronic structures as well as charge/spin transport properties are directly probed using state-of-art techniques.