Empowered by ever-increasing computational power and algorithmic developments, electronic-structure simulations continue to drive research and innovation in materials science. In this context, ab-initio calculations offer an unbiased platform for the understanding, development, design, and discovery of materials. The treatment of the electron interactions is at the core of any ab-initio method, with the accuracy of its solution affecting the final result. Here, dynamical functionals can play a crucial role in allowing the accurate prediction of spectral and thermodynamic properties of materials. In this thesis we develop a framework to deal with dynamical quantities, and a novel dynamical functional to address the electronic structure of correlated materials. We design the so-called sum-over-poles representation for dynamical propagators to perform accurately the calculations in dynamical frameworks. Then, we draw a link between the funda- mental equation of Green’s function formalism, the Dyson equation, and nonlinear eigenvalue problems, also highlighting that the Dyson equation is the (nonlinear) generalization of the Schrödinger equation for embedded systems. Notably, the sum-over-poles representation of the dynamical potential allows for an exact solution of the nonlinear problem by mapping the interacting system to a non-interacting “fictitious” system with augmented degrees of freedom and having the same Green’s function of the interacting system, with the spurious degrees of freedom traced away. The (linear) diagonalization of the effective Hamiltonian for the “fictitious” system yields the Dyson orbitals of the material, as a solution of the nonlinear problem, and its excitation energies as poles of the Green’s function. Also, the sum-over-poles representation of the Green’s function is known and allows for the computation of accurate spectroscopic and thermodynamic quantities. Furthermore, we introduce a novel approximation to the exchange-correlation part of the Luttinger-Ward functional, that generalizes the energy functional of DFT+U to host a dy- namically screened potential U(ω). Exploiting a localized-GW approach, we combine the precision of DFT+U for ground-state properties with the accuracy of GW for spectroscopic quantities and design the so-called dynamical Hubbard (Luttinger-Ward) functional. This yields a localized-GW self-energy as derivative, and simplifies to DFT+U in the case of a static screening. To test the approach, we use the algorithmic-inversion method on sum over poles to calculate the spectroscopic and thermodynamic quantities of the homogeneous electron gas at the GW level, finding very good agreement with previous results. Finally, we combine the algorithmic-inversion method on sum over poles with the dynamical Hubbard functional, to study the electronic structure of correlated materials. We apply the framework to compute the spectral, thermodynamic, and vibrational properties of SrVO3, finding results in excellent agreement with experiments and state-of-the-art computational methods, at a negligible computational cost.