We report on the propagation dynamics of Airy light beams under nonparaxial conditions. The partial waves forming the Airy beam can be divided into two parts, the first of which contains only propagating waves, while the second part consists of evanescent waves. In this Letter we propose the concept of the evanescent Airy beam. We analyze the structure of the ideal evanescent Airy beam, the initial profile of which has the Airy form, while its spectral decomposition consists of only evanescent partial waves. Also, we discuss the refraction of the Airy beam through an interface and investigate the field of the transmitted Airy beam.