  We consider an inflationary scenario where the rate of inflaton roll defined
by $\ddot\phi/H\dot \phi$ remains constant. The rate of roll is small for
slow-roll inflation, while a generic rate of roll leads to the interesting case
of `constant-roll' inflation. We find a general exact solution for the inflaton
potential required for such inflaton behaviour. In this model, due to non-slow
evolution of background, the would-be decaying mode of linear scalar
(curvature) perturbations may not be neglected. It can even grow for some
values of the model parameter, while the other mode always remains constant.
However, this always occurs for unstable solutions which are not attractors for
the given potential. The most interesting particular cases of constant-roll
inflation remaining viable with the most recent observational data are
quadratic hilltop inflation (with cutoff) and natural inflation (with an
additional negative cosmological constant). In these cases even-order slow-roll
parameters approach non-negligible constants while the odd ones are
asymptotically vanishing in the quasi-de Sitter regime.
