Fluid turbulence has made significant progress in recent years, with the application of statistical mechanics to Lagrangian dynamics leading to new quantitative theories of intermittency. The first analytical description of anomalous scaling laws in turbulence was achieved, revealing the role of statistical integrals of motion in non-equilibrium systems. These conserved quantities, hidden in the evolution of fluid particle groups, arise from the competition between expansion and geometry change, breaking scale invariance and causing observed anomalous scaling. Lagrangian methods also provide insights into practical issues like mixing and magnetic dynamos. The review focuses on fluid turbulence from a Lagrangian perspective, analyzing particle dynamics and their implications for transported fields. It discusses single-particle diffusion, two-particle dispersion in smooth and nonsmooth flows, multiparticle dynamics, and statistical conservation laws. Passive scalar and vector fields are analyzed, including their cascades and statistics. The review also covers the Burgers and Navier-Stokes equations, highlighting the role of Lagrangian dynamics in understanding turbulence. Key findings include the role of statistical integrals of motion in explaining intermittency and anomalous scaling, the breakdown of scale invariance in turbulent flows, and the importance of Lagrangian methods in understanding fluid dynamics. The review emphasizes the universal aspects of inertial interval statistics and the impact of finite injection scales on turbulence. It concludes with the significance of these findings for future research in turbulence.Fluid turbulence has made significant progress in recent years, with the application of statistical mechanics to Lagrangian dynamics leading to new quantitative theories of intermittency. The first analytical description of anomalous scaling laws in turbulence was achieved, revealing the role of statistical integrals of motion in non-equilibrium systems. These conserved quantities, hidden in the evolution of fluid particle groups, arise from the competition between expansion and geometry change, breaking scale invariance and causing observed anomalous scaling. Lagrangian methods also provide insights into practical issues like mixing and magnetic dynamos. The review focuses on fluid turbulence from a Lagrangian perspective, analyzing particle dynamics and their implications for transported fields. It discusses single-particle diffusion, two-particle dispersion in smooth and nonsmooth flows, multiparticle dynamics, and statistical conservation laws. Passive scalar and vector fields are analyzed, including their cascades and statistics. The review also covers the Burgers and Navier-Stokes equations, highlighting the role of Lagrangian dynamics in understanding turbulence. Key findings include the role of statistical integrals of motion in explaining intermittency and anomalous scaling, the breakdown of scale invariance in turbulent flows, and the importance of Lagrangian methods in understanding fluid dynamics. The review emphasizes the universal aspects of inertial interval statistics and the impact of finite injection scales on turbulence. It concludes with the significance of these findings for future research in turbulence.