In this thesis, we focus on Lagrangian investigations of isotropic turbulence, wall-bounded turbulence and vortex dynamics. In particular, the evolutionary multi-scale geometry of Lagrangian structures is quantified and analyzed. Additionally, we also study the dynamics of vortex-surface fields for some simple viscous flows with both Taylor--Green and Kida--Pelz initial conditions. First, we study the non-local geometry of finite-sized Lagrangian structures in both stationary, evolving homogenous isotropic turbulence and also with a frozen turbulent velocity field. The multi-scale geometric analysis is applied on the evolution of Lagrangian fields, obtained by a particle-backward-tracking method, to extract Lagrangian structures at different length scales and to characterize their non-local geometry in a space of reduced geometrical parameters. Next, we report a geometric study of both evolving Lagrangian, and also instantaneous Eulerian structures in turbulent channel flow at low and moderate Reynolds numbers. A multi-scale and multi-directional analysis, based on the mirror-extended curvelet transform, is developed to quantify flow structure geometry including the averaged inclination and sweep angles of both classes of turbulent structures at multiple scales ranging from the half-height of the channel to several viscous length scales. Results for turbulent channel flow include the geometry of candidate quasi-streamwise vortices in the near-wall region, the structural evolution of near-wall vortices, and evidence for the existence and geometry of structure packets based on statistical inter-scale correlations. In order to explore the connection and corresponding representations between Lagrangian kinematics and vortex dynamics, we develop a theoretical formulation and numerical methods for computation of the evolution of a vortex-surface field. Iso-surfaces of the vortex-surface field define vortex surfaces. A systematic methodology is developed for constructing smooth vortex-surface fields for initial Taylor--Green and Kida--Pelz velocity fields by using an optimization approach. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Numerical results on the evolution of vortex-surface fields clarify the continuous vortex dynamics in viscous Taylor--Green and Kida--Pelz flows including the vortex reconnection, rolling-up of vortex tubes, vorticity intensification between anti-parallel vortex tubes, and vortex stretching and twisting. This suggests a possible scenario for explaining the transition from a smooth laminar flow to turbulent flow in terms of topology and geometry of vortex surfaces.