In order to reduce the carbon foot-print of turbo-generators to global warming, a step-change in the design process is needed. Extensive CFD simulations coupled with optimization algorithms are required, resorting to novel techniques to capture the complex flow phenomena occurring in the passage. However, the computation of the optimization gradients and the imposition of geometrical constrains is computationally expensive and non-trivial due to the large number of design variables. In the 1980s, the adjoint method arose as an alternative technique to compute the sensitivities at a cost independent of the number of design variables. This technique is extremely popular in external aerodynamic applications. However, its full potential is yet to be exploited to internal flows. The use of an adjoint solver, together with a surface parametrization technique based on traditional blade design parameters, presents a unique opportunity towards a fully-automated design methodology based on high-fidelity models for turbomachinery applications. Stemming from the above considerations, the aim of this thesis is to develop a common open-source numerical framework for the optimization of both axial and radial turbomachinery. In order to achieve so, the open-source CFD suite SU2, that includes an adjoint solver, is coupled with ParaBlade, an open-source blade parametrizer based on traditional turbomachinery design variables. The proposed methodology is successfully applied in an axial turbine stator, where a reduction of 7.59% in the entropy generation across the passage is achieved. The study performed in a mixed-flow turbine rotor highlights the need to parametrize the hub and shroud surfaces. However, its optimization revealed a reduction in the objective function value, showing the robustness of the proposed methodology in two types of turbomachinery.