In this letter, we propose a novel symbol-asynchronous non-orthogonal multiple access (ANOMA) scheme for downlink transmissions. First, ANOMA introduces an artificial symbol-offset between packets destined for different user equipment (UE) in order to reduce their mutual interference. Second, ANOMA adopts a pre-coding at the base station and a whitening-and-decomposing (WD) detection scheme at the receiving UE to simplify signal detection. With these two novel techniques, ANOMA can significantly enlarge the achievable rate region of the traditional non-orthogonal multiple access (NOMA). Moreover, the impact of pulse shape and symbol offset on the performance of ANOMA scheme are investigated in both Gaussian and Rayleigh block-fading channels. The numerical results show that ANOMA with precoding and WD detection outperforms asynchronous NOMA with successive interference cancellation detection, both outperform synchronous NOMA.