This article reviews recent experimental and theoretical progress on many-body phenomena in dilute, 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 in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near Feshbach resonances in the BCS-BEC crossover. The article covers topics including optical lattices, detection of correlations, many-body effects in optical lattices, cold gases in one dimension, two-dimensional Bose gases, Bose gases in fast rotation, BCS-BEC crossover, and perspectives. It discusses the role of Feshbach resonances in tuning interactions and the importance of optical potentials in generating strong periodic potentials for cold atoms. The review also highlights the significance of ultracold gases in studying quantum phenomena, such as superfluidity, superconductivity, and quantum magnetism, and their potential applications in quantum information and condensed matter physics. The article emphasizes the importance of understanding the many-body behavior of ultracold gases, which can provide insights into fundamental physics and enable the exploration of new quantum states of matter.This article reviews recent experimental and theoretical progress on many-body phenomena in dilute, 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 in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near Feshbach resonances in the BCS-BEC crossover. The article covers topics including optical lattices, detection of correlations, many-body effects in optical lattices, cold gases in one dimension, two-dimensional Bose gases, Bose gases in fast rotation, BCS-BEC crossover, and perspectives. It discusses the role of Feshbach resonances in tuning interactions and the importance of optical potentials in generating strong periodic potentials for cold atoms. The review also highlights the significance of ultracold gases in studying quantum phenomena, such as superfluidity, superconductivity, and quantum magnetism, and their potential applications in quantum information and condensed matter physics. The article emphasizes the importance of understanding the many-body behavior of ultracold gases, which can provide insights into fundamental physics and enable the exploration of new quantum states of matter.