In order to control the propagation trajectory of an Airy beam, generalized Airy beams are proposed theoretically. Such beams can be generated by modulating a Gaussian beam imposed on a tunable phase pattern, where two angle factors θ1 and θ2 are introduced in cubic phase. The hyperbolic parabola is investigated based on diffraction catastrophe theory. We focus on the nondiffracting property in the propagation distances of 0, 3, 6, 10cm in the case of θ1=π/3 andθ2=7π/6, and the selfhealing property in the case of θ1=3π/2 and θ2=π/4. The experiment and numerical simulations demonstrate that such beams still possess the nondiffracting, selfhealing properties to some extent, although the theory shows that they are not the solutions for paraxial wave equation. The generalized Airy beams can propagate along arbitrarily appointed parabolic trajectories and directions in free space, and their intensity distributions are different from that of traditional Airy beams, which make the generalized Airy beams have potential applications in optical trapping and biomedicine.