Nonlinear interactions of ion cyclotron range of frequency (ICRF) waves with fusion plasmas are reviewed. Although the linear theory of ICRF waves, including fast waves (FWs), high-harmonic fast waves and ion Bernstein waves (IBWs), is widely applicable, nonlinear effects can still be important, especially in the edge plasma or for novel core applications. Here the topics of flow drive, ponderomotive forces, radiofrequency (rf) sheaths, parametric decay and related interactions with the edge plasma are considered. Primary emphasis is placed on the basic underlying physics and tokamak applications. For FW antennas, the parallel electric field near launching structures is known to drive rf sheaths which can give rise to convective cells, interaction with plasma 'blobs', impurity production and edge power dissipation. In addition to sheaths, IBW waves in the edge plasma are subject to strong ponderomotive effects and parametric decay. In the core plasma, slow waves can sometimes induce nonlinear effects. Mechanisms by which these waves can influence the radial electric field and its shear are summarized and related to the general (reactive-ponderomotive and dissipative) force on a plasma from rf waves. Standard ICRF codes have begun to incorporate the nonlinear topics described here. Further progress in integrated simulation should allow new predictive modelling capabilities.