This article reviews recent experimental and theoretical progress in the field of many-body physics with ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, and lowest Landau level physics in quasi-two-dimensional gases under fast rotation. The strong correlations in fermionic gases are discussed in optical lattices or near Feshbach resonances, where the BCS-BEC crossover can be explored. The review covers the scattering of ultracold atoms, weak interactions, and Feshbach resonances, as well as the detection of correlations and the many-body effects in optical lattices. It also discusses the Bose-Hubbard model, superfluid-Mott-insulator transitions, and the dynamics near quantum phase transitions. The article further explores cold gases in one dimension, two-dimensional Bose gases, and Bose gases in fast rotation, including the Lowest Landau Level formalism and artificial gauge fields. The BCS-BEC crossover is examined, including molecular condensates, crossover theory, and experiments near the unitarity limit. Finally, the perspectives on future research directions, such as quantum magnetism, disorder, and nonequilibrium dynamics, are discussed.This article reviews recent experimental and theoretical progress in the field of many-body physics with ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, and lowest Landau level physics in quasi-two-dimensional gases under fast rotation. The strong correlations in fermionic gases are discussed in optical lattices or near Feshbach resonances, where the BCS-BEC crossover can be explored. The review covers the scattering of ultracold atoms, weak interactions, and Feshbach resonances, as well as the detection of correlations and the many-body effects in optical lattices. It also discusses the Bose-Hubbard model, superfluid-Mott-insulator transitions, and the dynamics near quantum phase transitions. The article further explores cold gases in one dimension, two-dimensional Bose gases, and Bose gases in fast rotation, including the Lowest Landau Level formalism and artificial gauge fields. The BCS-BEC crossover is examined, including molecular condensates, crossover theory, and experiments near the unitarity limit. Finally, the perspectives on future research directions, such as quantum magnetism, disorder, and nonequilibrium dynamics, are discussed.