Grégoire Allaire's paper introduces the concept of "two-scale convergence," which is designed to better describe sequences of oscillating functions. The paper proves that bounded sequences in \( L^2(\Omega) \) are relatively compact with respect to this new type of convergence. A corrector-type theorem is also established, allowing for the replacement of a sequence by its "two-scale" limit, up to a strongly convergent remainder in \( L^2(\Omega) \). These results are particularly useful for the homogenization of partial differential equations with periodically oscillating coefficients. The paper proposes a new method for proving the convergence of homogenization processes, which is an alternative to the energy method of Tartar. The power and simplicity of the two-scale convergence method are demonstrated through several examples, including the homogenization of both linear and nonlinear second-order elliptic equations. The key to the method is the use of test functions that are periodic in one variable and smooth in the other, leading to a well-posed system of equations that combines the usual homogenized and cell equations. This system simplifies the structure of the limit problem while eliminating "strange" effects like memory or nonlocal effects that can arise in the usual homogenized problem.Grégoire Allaire's paper introduces the concept of "two-scale convergence," which is designed to better describe sequences of oscillating functions. The paper proves that bounded sequences in \( L^2(\Omega) \) are relatively compact with respect to this new type of convergence. A corrector-type theorem is also established, allowing for the replacement of a sequence by its "two-scale" limit, up to a strongly convergent remainder in \( L^2(\Omega) \). These results are particularly useful for the homogenization of partial differential equations with periodically oscillating coefficients. The paper proposes a new method for proving the convergence of homogenization processes, which is an alternative to the energy method of Tartar. The power and simplicity of the two-scale convergence method are demonstrated through several examples, including the homogenization of both linear and nonlinear second-order elliptic equations. The key to the method is the use of test functions that are periodic in one variable and smooth in the other, leading to a well-posed system of equations that combines the usual homogenized and cell equations. This system simplifies the structure of the limit problem while eliminating "strange" effects like memory or nonlocal effects that can arise in the usual homogenized problem.