The standard way of solving nonlinear Schrödinger equations (NLS) is to rewrite the differential equations into the equivalent form of integral equations, and then apply Picard iteration. In this process the Strichartz estimate is usually used to control the nonlinear term. When we consider the periodic NLS, however, the exact periodic analogue of the continuous Strichartz estimate fails, so it forces us to find some new inequalities of the same type. The periodic Strichartz estimate is also equivalent to the discrete Fourier restriction estimate. The results for this type of restriction as well as some other related topics will be presented.