The Thesis proposes a new theoretical framework for the study of electronic stopping of particle projectiles in crystalline solids. It does not rely on perturbative or linear response approximations. Moreover, it goes beyond nonlinear models for the homogeneous electron gas, which assume ideal metal hosts. The theory exploits a discrete symmetry in the trajectory of the projectile following a direction of crystalline periodicity, which allows a treatment based on Floquet theory for time-periodic systems. Floquet theory allows to find the solutions of time-periodic Hamiltonians through the same techniques used in time-independent problems: this provides the new framework with an intrinsic advantage over the the real time first principles calculations that are currently employed for analysing electronic stopping, which rely on the explicit solution of the time-dependent Schr"odinger equation, and are therefore very computationally expensive. The (stroboscopic) stationary solutions of the stopping problem are found using a Bloch-Floquet scattering treatment. The expressions for electronic stopping of previous perturbative and nonlinear models are readily recovered from the theory in the corresponding limits. The so-called “threshold velocity effect” for stopping is analysed and interpreted using quasienergy conservation, and it is suggested to display a much richer behaviour compared to both experimental observations (due to limited resolution), and previous phenomenological theoretical explanations. A method for numerical calculations is proposed: it is based on a tight-binding model, in which a time-evolving localised basis set is introduced to allow the treatment of the moving crystal in the projectile frame of reference and a Dyson equation can be solved for the Green’s function in the local basis set. While the model is presented for the one-band tight-binding model for simplicity, it can be generalised to higher dimensions and to arbitrary number of basis states per unit cell. In addition, other fundamental questions on this paradigmatic nonequilibrium problem, such as the adiabatic limit for a slow projectile, are discussed.