This review article discusses recent advancements in the physics of ultracold atomic and molecular gases in optical lattices, highlighting their potential to mimic condensed matter phenomena. The authors explore how these systems can serve as "quantum simulators" to address challenging questions in condensed matter physics and high energy physics. The review covers various topics, including: 1. **Introduction**: Provides a historical perspective on cold atom physics and its challenges in condensed matter physics. 2. **The Hubbard and Spin Models with Ultracold Lattice Gases**: Discusses the realization of Hubbard models and spin models in ultracold atomic gases. 3. **Methods of Treatment**: Reviews theoretical methods for treating Hubbard models, including weak and strong interaction limits, mean-field approaches, exact diagonalizations, quantum Monte Carlo, and phase space methods. 4. **Disordered Ultracold Atomic Gases**: Explores the realization of disorder in ultracold atomic gases and its effects on quantum phases. 5. **Frustrated Models in Cold Atom Systems**: Investigates frustrated models such as quantum antiferromagnets and their realization in cold atom systems. 6. **Ultracold Spinor Atomic Gases**: Examines spinor interactions and the realization of spinor gases in optical lattices. 7. **Ultracold Atomic Gases in "Artificial" Magnetic Fields**: Discusses the creation of artificial magnetic fields and their impact on ultracold gases. 8. **Quantum Information with Ultracold Gases**: Explores the connection between ultracold gases and quantum information, including entanglement and quantum computing. The review aims to bridge the gap between atomic physics, quantum optics, and condensed matter physics, providing a comprehensive overview of the current state of research and future prospects.This review article discusses recent advancements in the physics of ultracold atomic and molecular gases in optical lattices, highlighting their potential to mimic condensed matter phenomena. The authors explore how these systems can serve as "quantum simulators" to address challenging questions in condensed matter physics and high energy physics. The review covers various topics, including: 1. **Introduction**: Provides a historical perspective on cold atom physics and its challenges in condensed matter physics. 2. **The Hubbard and Spin Models with Ultracold Lattice Gases**: Discusses the realization of Hubbard models and spin models in ultracold atomic gases. 3. **Methods of Treatment**: Reviews theoretical methods for treating Hubbard models, including weak and strong interaction limits, mean-field approaches, exact diagonalizations, quantum Monte Carlo, and phase space methods. 4. **Disordered Ultracold Atomic Gases**: Explores the realization of disorder in ultracold atomic gases and its effects on quantum phases. 5. **Frustrated Models in Cold Atom Systems**: Investigates frustrated models such as quantum antiferromagnets and their realization in cold atom systems. 6. **Ultracold Spinor Atomic Gases**: Examines spinor interactions and the realization of spinor gases in optical lattices. 7. **Ultracold Atomic Gases in "Artificial" Magnetic Fields**: Discusses the creation of artificial magnetic fields and their impact on ultracold gases. 8. **Quantum Information with Ultracold Gases**: Explores the connection between ultracold gases and quantum information, including entanglement and quantum computing. The review aims to bridge the gap between atomic physics, quantum optics, and condensed matter physics, providing a comprehensive overview of the current state of research and future prospects.