Recently, topological classifications of quantum phases have been extended from non-interacting insulators to strongly correlated insulators, and further to semimetals. Recent results on direct visualizations of several topological quantum states with angle-resolved photoemission spectroscopy (ARPES). The direct observation the metallic surface states on strongly correlated Kondo insulator SmB6 and its helical spin texture of as compelling evidences for the predicted topological kondo insulator. Direct observation of 3D Weyl cones in the bulk states of topological semimetal TaAs, as experimental evidence of Weyl semimetal states. Discovery simple Weyl semimetal TaP, where only single type of Weyl fermions contributing the exotic transport properties. Observation of Fermi arc states in MoTe2 as evidence of type-II Weyl semimetal state. Topological insulators and topological semimetals are both new classes of quantum materials, which are characterized by surface states induced by the topology of the bulk band structure.