  We generalize the notion of constant-roll inflation earlier introduced in
General Relativity (GR) and $f(R)$ gravity to inflationary models in more
general scalar-tensor gravity. A number of novel exact analytic solutions for a
FLRW spatially flat cosmological background is found for this case. All forms
of the scalar field potential and its coupling to gravity producing the exact
de Sitter solution, while the scalar field is varying, are presented. In the
particular cases of induced gravity and GR with a non-minimally coupled scalar
field, all constant-roll inflationary solutions are found. In the former case
they represent power-law inflation, while in the latter case the solution is
novel and more complicated. Comparison of scalar perturbations generated during
such inflation in induced gravity with observational data shows that the
constant-roll parameter should be small, similar to constant-roll inflation in
GR and $f(R)$ gravity. Then the solution reduces to the standard slow-roll one
with small corrections.
