Abstract: This paper mainly studies both the phase and frequency synchronization of discrete-time Kuramoto oscillators with phase shift. Firstly, a sufficient condition for achieving phase synchronization is established by analyzing the condition under which the maximum phase difference of any two oscillators at the same time tends to be zero. Secondly, when the intrinsic frequencies of the oscillators are nonidentical to each other, a useful bounded property of the phase difference is obtained and then some sufficient conditions for achieving frequency synchronization are derived by setting the upper limit of the phase difference between any two oscillators at the initial time. Finally, the numerical simulations of a system with ten oscillators further show that, when the frequencies of the oscillators are identical to each other, the difference between the maximum and the minimum phases can reduce to 0.003 6 when t=20 s, while, when the frequencies are nonidentical to each other, the difference between the maximum and the minimum frequencies of each oscillator can even reach 4.066×10-5 when t=0.3 s. The numerical simulations further verify the correctness of the theoretical results.