This integrated thesis documents a series of five complementary numerical investigations aimed at understanding the flow instability and laminar-to-turbulent transition process in attached and separated shear layers. Direct numerical simulation was used in these studies to accurately resolve the spatial and temporal scales of the simulated flows. The first two investigations simulate the flow in an idealized computational domain that approximates the conditions on the suction surface of a low-pressure turbine (LPT) blade. The first study investigates the interaction of the Tollmien-Schlichting instability in the attached-flow region with the Kelvin-Helmholtz instability in the separated-flow region. The interaction produces packets of coherent vortices that facilitate transition to a fully-turbulent boundary layer. The effect of swept-blade conditions on the transition process is investigated in the second numerical study by sweeping the simulation domain by 45 degrees. The resulting three-dimensional pressure field creates the potential for a crossflow-induced instability mode, but its effect on the transition process is minimal and transition follows the same mechanisms observed in the unswept configuration. The characteristics of the coherent vortices that facilitate transition to turbulence in the first study are further investigated through a series of fundamental studies of an isolated turbulent spot that is triggered by a transverse jet. The first of these fundamental studies demonstrates the sensitivity of the formation and structure of the coherent vortices to the level of free-stream acceleration. A subsequent fundamental study identifies the regeneration process by which the turbulent spot grows laterally through the formation of packets of hairpin vortices along the spanwise edges of the spot and longitudinally by the increase in the spatial scale of the hairpin vortices. The final fundamental study investigates the effects of elevated free-stream turbulence levels and LPT-realistic pressure distributions on the transition process. Under elevated turbulence conditions, bypass transition occurs as streamwise streaks form in the laminar boundary layer and roll-up into hairpin-shaped vortices via a Kelvin-Helmholtz instability mode that is accelerated by the free-stream turbulence. The wave-packet regeneration model identified under low free-stream turbulence is promoted by elevated free-stream turbulence, increasing the growth-rate of turbulent spots under elevated turbulence conditions.