We present theoretically and experimentally generalized and symmetric Airy beams, where the two sidelobes are not mutually perpendicular, by introducing two rotary angle factors. The symmetric Airy beam is induced by a binary phase pattern. We demonstrate that the intensity distributions of generalized Airy beams are apparently different from those of normal Airy beams. Moreover, they can propagate along arbitrary trajectories. Numerical results show that the generalized and symmetric Airy beams still have the ability of self-healing and nondiffraction. The experimental results are in complete accord with numerical results. Some possible applications are also discussed, and these interesting properties will also likely have potential applications.