Uniqueness Questions for Multiple Trigonometric Series

We survey some recent results on the uniqueness questions on multiple trigonometric series. Two basic questions, one about series which converges to zero and the other about the series which converge to an integrable function, are asked for four modes of convergence: unrestricted rectangular convergence, spherical convergence, square convergence, and restricted rectangular convergence. We will either get into the details or outline some of the proofs for the known uniqueness theorems. Some results on the sets of uniqueness are also given. Finally, we will mention some interesting open questions in this area. Some of them are even one-dimensional. We assume the reader has some basic knowledge of measure theory and Fourier analysis. Most of the topics and materials can be understood by upper level undergraduate students.