  Strongly non-geodesic, or rapidly turning trajectories in multifield
inflation have attracted much interest recently from both theoretical and
phenomenological perspectives. Most models with large turning rates in the
literature are formulated as effective field theories. In this paper we
investigate rapid-turn inflation in supergravity as a first step towards
understanding them in string theory. We find that large turning rates can be
generated in a wide class of models, at the cost of high field space curvature.
In these models, while the inflationary trajectories are stable, one Hessian
eigenvalue is always tachyonic and large, in Hubble units. Thus, these models
satisfy the de Sitter swampland conjecture along the inflationary trajectory.
However, the high curvatures underscore the difficulty of obtaining rapid-turn
inflation in realistic string-theoretical models. In passing, we revisit the
$\eta$-problem in multifield slow-roll inflation and show that it does not
arise, inasmuch as the inflatons, $\phi^i$, can all be heavier (in absolute
value) that the Hubble scale: $|m_i|/H>1$, $\forall\, i$.
