In this work, the boundary-layer stability of a blunted cone in different atmospheres and under different free-stream Reynolds numbers will be investigated to examine the influence of the Mars atmosphere on the laminar-turbulent transition. Therefore, a blunted cone, experimentally investigated in a pure CO₂ atmosphere to measure the wall heat flux distribution, was chosen from literature and the linear stability theory and the linear parabolized stability equations were applied for the investigations. To perform the investigations, firstly, a new Navier-Stokes solver, named CONSST3D, to calculate the laminar base-flow solution and a new boundary-layer stability solver, named COSTAS, were developed and will be presented in this work containing the governing equations, the thermodynamic models and the numerical implementation. To validate the two solvers for the perfect gas and the thermo-chemical equilibrium gas regime, calculations were performed for the Stetson Mach 8 blunted cone and a Mach 10 flat plate, respectively. With the successfully validated solvers, the boundary-layer stability calculations were performed for the blunted cone in a pure CO₂ atmosphere, in the Mars atmosphere and in the Earth atmosphere requiring similar free-stream conditions to examine the influence of the atmosphere on the laminar-turbulent transition and to get first insights in the boundary-layer stability of a blunted cone in the Mars atmosphere. The results showed a destabilizing effect of the CO₂ atmosphere and the Mars atmosphere compared to the Earth atmosphere by applying a thermo-chemical equilibrium gas model. Further, the transition onset N-factor was found to be in the typical range of values at the transition onset location for the test case in the CO₂ atmosphere. The comparison with the calculation in the Mars atmosphere showed only small differences in the boundary-layer stability and thus, test campaigns in a pure CO₂ atmosphere were found to be a good approximation of the Mars atmosphere which contain a small additional amount of N₂. Overall, in comparison to the calculations in the Earth atmosphere, higher growth rates and higher N-factors were found in the CO₂ and Mars atmospheres at lower disturbance frequencies. An additional calculation was performed in the CO₂ atmosphere at a different free-stream condition, where the Reynolds number was significantly higher compared to the previous test case. In this case, the results showed a significantly higher N-factor at the transition onset location, which is located further upstream compared to the lower free-stream condition test case.