This paper by R.J. DiPerna and P.L. Lions presents new existence, uniqueness, and stability results for ordinary differential equations (ODEs) with coefficients in Sobolev spaces. These results are derived from corresponding results on linear transport equations, analyzed using the method of renormalized solutions. The paper begins with an introduction to the Cauchy-Lipschitz theorem, which provides global solutions for ODEs with Lipschitz continuous coefficients. It then discusses linear transport equations, focusing on existence, uniqueness, and stability. The paper also explores the concept of renormalized solutions and their duality. Applications of these results to ODEs are presented, including autonomous and time-dependent cases. The paper also includes counterexamples and remarks, highlighting the limitations of Sobolev vector-fields, such as those with unbounded divergence or without integrable first derivatives. The authors also discuss small noise approximations and other related remarks. The paper emphasizes the importance of measure theory in extending classical results to Sobolev spaces. A key result is the estimation of the Lebesgue measure under the flow of an ODE, which is derived from the transport equation. The paper provides a detailed analysis of this result, showing that the measure evolves according to a transport equation and admits a density that satisfies a related equation. The paper concludes with a discussion of the implications of these results for the study of ODEs with Sobolev coefficients.This paper by R.J. DiPerna and P.L. Lions presents new existence, uniqueness, and stability results for ordinary differential equations (ODEs) with coefficients in Sobolev spaces. These results are derived from corresponding results on linear transport equations, analyzed using the method of renormalized solutions. The paper begins with an introduction to the Cauchy-Lipschitz theorem, which provides global solutions for ODEs with Lipschitz continuous coefficients. It then discusses linear transport equations, focusing on existence, uniqueness, and stability. The paper also explores the concept of renormalized solutions and their duality. Applications of these results to ODEs are presented, including autonomous and time-dependent cases. The paper also includes counterexamples and remarks, highlighting the limitations of Sobolev vector-fields, such as those with unbounded divergence or without integrable first derivatives. The authors also discuss small noise approximations and other related remarks. The paper emphasizes the importance of measure theory in extending classical results to Sobolev spaces. A key result is the estimation of the Lebesgue measure under the flow of an ODE, which is derived from the transport equation. The paper provides a detailed analysis of this result, showing that the measure evolves according to a transport equation and admits a density that satisfies a related equation. The paper concludes with a discussion of the implications of these results for the study of ODEs with Sobolev coefficients.