This work provides a detailed investigation of matrix product state (MPS) representations for pure multipartite quantum states. It determines the freedom in representations with and without translation symmetry, derives canonical forms, and provides efficient methods for obtaining them. The results extend to frustration-free Hamiltonians and the generation of MPS, and discuss the use of MPS representations for classical simulations of quantum systems. The paper covers definitions, preliminaries, and various aspects of MPS, including their relation to valence bond pictures, finitely correlated states, and frustration-free Hamiltonians. It also explores the canonical form of MPS, the construction of local Hamiltonians with MPS as exact ground states, and the connection between MPS and sequential generation of multipartite entangled states. Additionally, it reviews how MPS efficiently approximate important states in nature, such as ground states of 1D local Hamiltonians, and discusses the role of entanglement in quantum computing.This work provides a detailed investigation of matrix product state (MPS) representations for pure multipartite quantum states. It determines the freedom in representations with and without translation symmetry, derives canonical forms, and provides efficient methods for obtaining them. The results extend to frustration-free Hamiltonians and the generation of MPS, and discuss the use of MPS representations for classical simulations of quantum systems. The paper covers definitions, preliminaries, and various aspects of MPS, including their relation to valence bond pictures, finitely correlated states, and frustration-free Hamiltonians. It also explores the canonical form of MPS, the construction of local Hamiltonians with MPS as exact ground states, and the connection between MPS and sequential generation of multipartite entangled states. Additionally, it reviews how MPS efficiently approximate important states in nature, such as ground states of 1D local Hamiltonians, and discusses the role of entanglement in quantum computing.