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ii Ultracold atoms have revolutionized the eld of quantum many-body physics due to excellent understanding of their microscopic dynamics and a high degree of control over these dynamics with external elds. The next revolution in ultracold physics promises to come with ultracold molecules, whose production lies at the cutting edge of research. In this thesis we are concerned with how ultracold molecules trapped in optical lattices may be used as resources for novel many-body physics. There are six main parts to this thesis. The rst part gives a general introduc-tion to the topics covered in the thesis, and provides some technical details on the derivation of many-body lattice models from few-body physics. In the second part, we derive a low-energy Hamiltonian, the molecular Hubbard Hamiltonian, describing ultracold heteronuclear bialkali dimer molecules loaded into an optical lattice and elucidate its many-body properties. These molecules have large permanent electric dipole moments which give rise to long range and anisotropic dipole-dipole interac