This dissertation is a comprehensive account of the low-Reynolds number (Re) flow over a cambered airfoil for a wide range of angles of attack with a focus on the dynamics of boundary layer separation and transition. The unsteady and complex phe- nomena of the transitional flow are analyzed through a combination of direct numerical simulations (DNS), large-eddy simulations (LES), experiments, and development of Lagrangian theory and methods. A discontinuous Galerkin spectral element method (DGSEM) is used to model the compressible Navier-Stokes equations in two and three dimensions. The DGSEM generates high-order accurate results with low dispersion and diffusion errors and has been developed to include kinetic-energy conserving volume fluxes, tools to efficiently track Lagrangian fluid tracers, and computation of higher wall-normal velocity derivatives. The code is benchmarked through a series of Navier-Stokes flows using different DG variants and polynomial orders. High-fidelity DNS in three dimensions show that the transitional flow over a cambered NACA 65(1)-412 airfoil at Re = 2×104 swiftly changes from a state of laminar separation at mid-chord without reattachment to a laminar separation bubble (LSB) at the leading edge with a turbulent boundary layer. The bifurcation occurs within an angle-of-attack change of two degrees and is accompanied by a rapid increase of the lift and decrease of the drag force, which is observed in computations and experiments likewise. Each flow regime is governed by different dynamics, instabilities, and wake structures that change with the transition location of the separated shear layer. The kinematic aspects of flow separation are further investigated in the Lagrangian frame, where the initial motion of upwelling fluid material from the wall is related to the long-term attracting manifolds in the flow field. An objective finite-time diagnostic for instabilities in shear flows based on the curvature of Lagrangian material lines is introduced. By defining a flow instability in the Lagrangian frame as the increased folding of lines of fluid particles, subtle perturbations and unstable growth thereof are detected early based solely on the curvature change of material lines over finite time.