The paper reviews the theory of Bose-Einstein condensation (BEC) in trapped gases, focusing on the role of interactions and the mean-field approach. It discusses the properties of trapped Bose gases, including density profiles, energy, collective oscillations, and dynamics of expansion. The thermodynamic limit exhibits scaling behavior in relevant length and energy scales. Despite the dilute nature of the gases, interactions significantly modify static and dynamic properties, with mean-field theory agreeing well with experimental results. The review also covers superfluidity, quantized vortices, and coherence effects like the Josephson effect and interference. The paper assesses the accuracy and limitations of the mean-field approach. The review begins with an introduction to BEC, its discovery in 1995, and its theoretical roots in quantum statistical mechanics. It discusses the experimental and theoretical research on BEC, including its connection to superfluidity in helium and other systems. The paper highlights the importance of inhomogeneity in trapped Bose gases and the role of two-body interactions. It discusses the application of mean-field theory, particularly the Gross-Pitaevskii theory, to describe BEC in harmonic traps. The paper also covers the thermodynamic properties of these systems, including the critical temperature, condensate fraction, and energy. It discusses the effects of interactions on the ground state, dynamic behavior, and thermal effects. The paper also addresses the role of dimensionality and the behavior of BEC in different dimensions. It discusses the effects of non-harmonic traps and adiabatic transformations. The paper concludes with a discussion of the future perspectives in the field of BEC.The paper reviews the theory of Bose-Einstein condensation (BEC) in trapped gases, focusing on the role of interactions and the mean-field approach. It discusses the properties of trapped Bose gases, including density profiles, energy, collective oscillations, and dynamics of expansion. The thermodynamic limit exhibits scaling behavior in relevant length and energy scales. Despite the dilute nature of the gases, interactions significantly modify static and dynamic properties, with mean-field theory agreeing well with experimental results. The review also covers superfluidity, quantized vortices, and coherence effects like the Josephson effect and interference. The paper assesses the accuracy and limitations of the mean-field approach. The review begins with an introduction to BEC, its discovery in 1995, and its theoretical roots in quantum statistical mechanics. It discusses the experimental and theoretical research on BEC, including its connection to superfluidity in helium and other systems. The paper highlights the importance of inhomogeneity in trapped Bose gases and the role of two-body interactions. It discusses the application of mean-field theory, particularly the Gross-Pitaevskii theory, to describe BEC in harmonic traps. The paper also covers the thermodynamic properties of these systems, including the critical temperature, condensate fraction, and energy. It discusses the effects of interactions on the ground state, dynamic behavior, and thermal effects. The paper also addresses the role of dimensionality and the behavior of BEC in different dimensions. It discusses the effects of non-harmonic traps and adiabatic transformations. The paper concludes with a discussion of the future perspectives in the field of BEC.