The Sun is a dynamic star made of plasma with a magnetized atmosphere, where mag- netohydrodynamic (MHD) waves are frequently observed. In this Thesis, the nonlinear evolution of standing torsional Alfv ́en waves in solar coronal structures, such as coronal loops, prominence threads, and coronal flux ropes, is investigated using three-dimensional numerical simulations. The open-source PLUTO code is used, which solves the nonlinear MHD equations using a finite volume formulation and implements the Adaptive Mesh Refinement technique. A coronal loop and a prominence thread are modeled as straight flux tubes filled with plasma that is denser than their environment. In turn, a coronal flux rope is modeled as a twisted magnetic field embedded in a uniform coronal plasma. The e↵ect of the solar phostosphere is included in the models through the line-tying boundary condition at the feet of the structures. Standing torsional Alfv ́en waves are excited by perturbing the component of velocity perpendicular to the magnetic field lines. Owing to the spatially-varying Alfv ́en frequency across the flux tube, caused by the nonuniformity of density and/or magnetic field, Alfv ́en waves oscillate independently from each other in adjacent magnetic surfaces. As a result, such waves develop phase mixing, which generates shear flows and transports the wave en- ergy towards larger and larger perpendicular wavenumbers as time increases. In this initial phase, the dynamics is quasi-linear. Simultaneously, other MHD waves can appear during the evolution due to either linear or nonlinear coupling. Slow magnetoacoustic waves are nonlinearly generated due to the ponderomotive force, while fast magnetoacoustic sausage waves are linearly generated if magnetic twist is present. Eventually, the phase-mixing shear flows trigger the Kelvin-Helmholtz instability (KHi), whose onset is unavoidable in flux tubes with a straight magnetic field. In weakly twisted tubes, the onset of the KHi is delayed, bu
