In this work, we investigate the effect of time asynchrony in non-orthogonal multiple access (NOMA) schemes for downlink transmissions. First, we analyze the benefit of adding intentional timing offsets to the conventional power domain-NOMA (P-NOMA). This method which is called Asynchronous-Power Domain-NOMA (AP-NOMA) introduces artificial symbol-offsets between packets destined for different users. It reduces the mutual interference which results in enlarging the achievable rate-region of the conventional P-NOMA. Then, we propose a precoding scheme which fully exploits the degrees of freedom provided by the time asynchrony. We call this multiple access scheme T-NOMA which provides higher degrees of freedom for users compared to the conventional P-NOMA or even the modified AP-NOMA. T-NOMA adopts a precoding at the base station and a linear preprocessing scheme at the receiving user which decomposes the broadcast channel into parallel channels circumventing the need for Successive Interference Cancellation (SIC). The numerical results show that T-NOMA outperforms AP-NOMA and both outperform the conventional P-NOMA. We also compare the maximum sum-rate and fairness provided by these methods. Moreover, the impact of pulse shape and symbol offset on the performance of AP-NOMA and T-NOMA schemes are investigated.