Quantum Many Body Physics of Ultracold Molecules in Optical Lattices : Models and Simulation Methods This thesis investigates ultracold molecules as a resource for novel quantum many body physics in particular by utilizing their rich internal structure and strong long range dipole dipole interactions. In addition numerical methods based on matrix product states are analyzed in detail and general algorithms for investigating the static and dynamic properties of essentially arbitrary one dimensional quantum many body systems are put forth. Finally this thesis covers open source implementations of matrix product state algorithms as well as educational material designed to aid in the use of understanding such methods