These lectures introduce aspects of gauge-string duality and holographic duality between a quantum system on the D+1 dimensional bulk and its dual system on the D dimensional boundary. Its mathematical and practical perspectives will be discussed. Gauge theory and string theory are two different theories defined in different ways, yet they have an equivalence in a nontrivial way. We describe the ‘t Hooft large N expansion in large N gauge theories. We then discuss Anti de Sitter Space/Conformal Field Theory correspondence and the GKPW relation. The three-point correlation functions related to the amplitudes on the gravity will be discussed. Other aspects of the holography duality on Wilson loops, renormalization groups, QCD, moduli spaces, Einstein manifolds, topological strings, AdS3 and AdS2, black holes, entanglement, and quantum information will be discussed in details, respectively. The Penrose limit of AdS space, the plane-wave geometries, the BMN correspondence, and the field theory operators with large scaling dimensions will also be discussed. The roles and aspects of permutation, Schur-Weyl duality, and nonplanarity in gauge theory and string theory will also be addressed.