Reflection and refraction of a finite-power Airy beam at the interface between two dielectric media are investigated analytically and numerically. The formulation takes into account the paraxial nature of the optical beams to derive convenient field evolution equations in coordinate frames moving along Snell’s refraction and reflection axes. Through numerical simulations, the self-accelerating dynamics of the Airy-like refracted and reflected beams are observed. Of special interest are the cases of critical incidence at Brewster and total-internal-reflection (TIR) angles. In the former case, we find that the reflected beam achieves to self-heal despite the severe supression of a part of its spectrum, while, in the latter case, the beam remains nearly unaffected except for the Goos-Hänchen shift. The self-accelerating quality persists even if the beam is trapped by multiple TIRs inside a dielectric film. Grazing incidence of an Airy beam at the interface between two media with close refractive indices is also investigated revealing that the interface can act as a filter cepending on the beam scale and tilt. We finally consider reverse refraction and perfect imaging of an Airy beam into a left-handed medium
