A new class of inflationary attractors characterized by a strongly non-geodesic motion has been discovered in the past few years. I describe how they naturally arise in negatively curved field space, allowing to inflate on potentials that are steep in Planck units, albeit without alleviating the ever-present naturalness issue of inflation. In these scenarios, fluctuations often experience a transient tachyonic instability, which can be described by a single-field effective field theory with an imaginary speed of sound. Independently of the precise ultraviolet origin of the latter, this leaves a peculiar imprint in the form of a high-level of primordial non-Gaussianities of flattened type for all higher-order correlation functions. On small scales, a transient phase of strongly non-geodesic motion provides a mechanism to generate primordial black holes and can leave specific signatures in the form of oscillations in the frequency profile of the stochastic gravitational wave background.