The chapter discusses the Lagrangian dynamics of fluid particles in turbulent flows, focusing on the behavior of single and multiple particles. It begins with the analysis of single-particle diffusion, where the motion is governed by a stochastic equation and exhibits either ballistic or diffusive behavior depending on the correlation time of the velocity field. The chapter then delves into the dispersion of two particles in a spatially smooth velocity field, where the separation between particles grows exponentially due to the Lagrangian strain matrix. The concept of Lyapunov exponents is introduced, which describe the asymptotic separation rate of particles. The chapter also explores the impact of compressibility on particle separation, leading to clustering rather than dispersion. Finally, it discusses the geometry and statistical conservation laws that emerge in multiparticle configurations, highlighting the role of statistical integrals of motion in understanding anomalous scaling and intermittency in turbulent transport.The chapter discusses the Lagrangian dynamics of fluid particles in turbulent flows, focusing on the behavior of single and multiple particles. It begins with the analysis of single-particle diffusion, where the motion is governed by a stochastic equation and exhibits either ballistic or diffusive behavior depending on the correlation time of the velocity field. The chapter then delves into the dispersion of two particles in a spatially smooth velocity field, where the separation between particles grows exponentially due to the Lagrangian strain matrix. The concept of Lyapunov exponents is introduced, which describe the asymptotic separation rate of particles. The chapter also explores the impact of compressibility on particle separation, leading to clustering rather than dispersion. Finally, it discusses the geometry and statistical conservation laws that emerge in multiparticle configurations, highlighting the role of statistical integrals of motion in understanding anomalous scaling and intermittency in turbulent transport.