The properties and possible triggering mechanisms of Alfvén waves in the reversed-field pinch (RFP) and circular tokamak configurations are discussed in the framework of nonlinear 3D magnetohydrodynamics (MHD) modeling. Numerical simulations are performed with the SpeCyl code (Cappello and Biskamp 1996 Nucl. Fusion) that solves the equations of the viscoresistive MHD model in cylindrical geometry. Configurations with increasing levels of complexity are analyzed. First, single-wave numerical solutions are compared with analytical ones in the simplest case of a uniform axial magnetic field: an excellent agreement is obtained for both the shear Alfvén wave (SAW) and the compressional Alfvén eigenmodes (CAEs). Then, tokamak and RFP configurations are studied. Phenomena such as phase mixing of SAW, resonant absorption of CAEs and the appearance of the global Alfvén eigenmode are described. Finally, the fully 3D RFP case with typical sawtoothing activity is investigated, showing for the first time in nonlinear RFP simulations the excitation of Alfvén waves by magnetic reconnection events. The modeling results appear to be consistent with the experimental characterization of Alfvénic activity observed in RFX-mod.