The goal of this ambitious project is to significantly contribute to the scientific program carried out for more than forty years –dating back to the work of Folland and Stein– in charge of extending the techniques from the Euclidean harmonic analysis to the more general setting of nilpotent Lie groups. My contribution will be the construction of the theory of Fourier Integral operators (FIOs) on graded Lie groups. An appropriate theory of FIOs in this setting will allow me to undertake the solution of many open problems of the non-commutative harmonic analysis. To achieve the goals of this project and for making tractable some of the aforementioned problems I will address the following tasks: (i) first, I will investigate the extension of the Euclidean Fourier integral operators theory due to Hörmander and Duistermaat to the setting of graded Lie groups. Here, I will use my expertise extending pseudo-differential theories as I did in my Ph.D. in the context of sub-Riemannian structures on compact Lie groups; (ii) I will study the continuity of the wave propagators for a Rockland operator R or for the power Rθ, 0 ≤ θ < 1, on Lebesgue spaces. For this, I will use the research experience that I collected in the last eight years in proving the boundedness properties of pseudo-differential operators and of Fourier integral operators; and (iii) I want to explore the validity of the controllability for parabolic and hyperbolic problems on domains of graded Lie groups.