The applicability of the quasilinear (QL) theory in the modelling of radio-frequency (RF) heating and current drive (CD) in nearly collisionless plasmas is investigated. In these experiments the wave spectrum excited by the generator and the antenna is normally coherent so that the standard justification of the QL theory, namely that particles interact with a random phase ensemble of waves, cannot be applied. The QL theory is, nevertheless, valid provided that the wave phase seen by nearly resonant particles is effectively randomized before the nonlinearity of the particle motion in the wave starts to become important. In the case of Landau resonances at frequencies in the ion cyclotron range or below, when electrons normally resonate with only one wave at a time, the main randomization mechanism is due to Coulomb collisions. A numerical technique to evaluate the diffusion coefficient of the standard map, due to Rechester et al (1981 Phys. Rev. A 23 2264), has been extended to take into account weak collisions. It allows an efficient and accurate evaluation of the velocity diffusion coefficient as a function of collisionality and wave amplitude, and, in particular, leads to an explicit quantitative criterion for the transition from the nonlinear to the QL regime.When this criterion is satisfied the heating rate predicted by the QL kinetic equation becomes independent of the collision frequency, and exactly balances the linear Landau damping rate of the wave. With a simplified but quantitatively correct model of the collision operator, it is shown that the same criterion justifies the linearization of the Vlasov equation to describe wave propagation. In the linear regime the two equations form a closed and internally consistent system for the description of RF heating and CD in fusion plasmas, with a well-defined validity criterion, which is satisfied by a large margin in most experimental situations.