This dissertation studies the effects of a discrete roughness element on a high-speed boundary layer using Direct Numerical Simulations (DNS) on unstructured grids. Flow past a cylindrical roughness element placed perpendicular to the flow and a hemispherical bump is studied. A compressible linear stability theory (LST) solver for parallel flows is developed based on the algorithm by Malik  and validated for a range of Mach numbers ranging from incompressible to Mach 10. The evolution of the perturbations from DNS is validated with the linear stability solver making the DNS algorithm suitable to study transition problems. Flow past a cylindrical roughness element at Mach 8.12 is simulated using DNS and the velocity profiles in the symmetry and wall--parallel planes are compared to the experiments of Bathel et al. . The flow remains steady and laminar, and does not transition. Overall, good agreement is observed between DNS and experiments, thus validating our algorithm to study effect of roughness on high-speed flows. However, differences are observed in the separation region upstream and recirculation region downstream of the roughness. The DNS results are used to quantify possible uncertainties in the measurement technique as suggested by Danehy . The effect of upstream injection (5% of the free-stream velocity) is also simulated to quantify its effects on the velocity profiles to mimic the injection of NO into air in the experiment. While the boundary layer thickness of the flow increases downstream of the injection location, its effect on the velocity profiles is small when the profiles are scaled with the boundary layer thickness.Flow past a hemispherical bump at Mach 3.37, 5.26 and 8.23 are simulated using DNS with the flow conditions matching the experiments of Danehy et al.  to understand the different flow features associated with the flow and the physical mechanism that causes the flow to transition to turbulence. It is observed that the Mach 3.37 and 5.26 flows transition to turbulence while the Mach 8.23 flow remains laminar downstream of the roughness element. The roughness element used in this study is large since the boundary layer thickness of the laminar boundary layer at the location of the roughness is smaller than the roughness height.The Mach 3.37 flow undergoes transition closer to the bump when compared to Mach 5.26, in agreement with experimental observations. Transition is accompanied by an increase in C_f and C_h (Stanton number). Even for the case that did not undergo transition (Mach 8.23), streamwise vortices induced by the roughness cause a significant rise in C_f until 20D downstream. Mean Van-Driest transformed velocity and Reynolds stress for Mach 3.37 and 5.26 shows good agreement with available data. The transition process involves the following key elements - Upon interaction with the roughness element, the boundary layer separates to form a series of spanwise vortices upstream of the roughness, and a separation shear layer. The system of spanwise vortices wrap around the roughness element in the form of horseshoe/necklace vortices to yield a system of counter-rotating streamwise vortices downstream of the element. These vortices are located beneath the separation shear layer and perturb it, which results in the formation of trains of hairpin-shaped vortices further downstream of the roughness for the cases that undergo transition. These hairpins spread in the span with increasing downstream distance and the flow increasingly resembles a fully developed turbulent boundary layer. A local Reynolds number based on the wall properties is seen to correlate the onset of transition for the cases considered.To assess the effect of roughness height on transition, a Mach 3.37 flow past a hemispherical bump is studied by varying the boundary layer thickness (k/delta = 2.54, 1.0, 0.25 & 0.125) where k is the roughness height and delta is the laminar boundary layer thickness at the location of the roughness. Transition occurs in all cases, and the essential mechanism of transition appears to be similar. At smaller boundary layer thickness, multiple trains of hairpin vortices are observed immediately downstream of the roughness, while a single train of hairpin vortices is observed at larger delta. This behavior is explained by the influence of the boundary layer thickness on the separation vortices upstream of the roughness element. Also, hairpin vortices that form downstream of the roughness initially scale with the height of the roughness element and further downstream, begin to scale with the boundary layer thickness, thus causing the entire boundary layer to transition. Dynamic Mode Decomposition of the pressure field for k/delta= 1 and 0.125 is used to obtain the frequency of shedding of hairpin vortices.