Launch vehicle structures are commonly composed of cylindrical and conical shells, which are inherently sensitive to buckling. The axial compression experienced during launch can consequently be a sizing load case, so it is important to understand the axial buckling behavior of these shells. Experimental testing is an essential part of studying this phenomenon because manufacturing imperfections can cause large discrepancies between theory and reality. In addition, conical shells have been researched less frequently than cylindrical shells, such that their behavior is less well understood. Experimental testing of launch vehicle structures is difficult and expensive due to their large size, hence it is preferred to test reduced-scale shells, representative of the full-scale ones. In this thesis, a scaling methodology is developed which allows designing representative reduced-scale conical shells for full-scale composite conical shells buckling in axial compression. The conical shells are assumed to have a symmetric, balanced layup with negligible flexural anisotropy. The scaling methodology is developed using the nondimensional governing equations, obtained through Nemeth's procedure, which allows to directly use the coefficients of the equations as scaling parameters. It also provides a framework to not only compare the buckling load of the shells of different sizes, but also the displacement upon buckling, the deformation shape, and the radial displacement. The methodology is set up such that the reduced-scale design parameters are determined sequentially. The buckling behavior of the two shells is compared using a semi-analytical approach, linear eigenvalue, and implicit dynamic finite element analyses. The eigenmode imperfection sensitivity is also evaluated. The methodology is successfully applied to isotropic, cross-ply, quasi-isotropic, and sandwich conical shells. The prediction accuracy is mainly affected by not being able to simultaneously satisfy all scaling parameters, by non-negligible flexural anisotropy and transverse shear, and by differences in imperfection sensitivity between the full-scale and reduced-scale shells. In any case, accurate results are obtained for the considered shells. The radial displacement is most difficult to predict, which is attributed to the membrane prebuckling assumption and neglecting the presence of imperfections. Finally, it is observed that larger eigenmode imperfections affect the accuracy, but they do not cause the methodology to fail. For future work, it is recommended to validate the methodology through experimental testing.