In this talk, we will consider the problem of multi-dimensional data completion. We will investigate the fundamental conditions on the sampling pattern, i.e., locations of the sampled entries, for completability of a low-rank tensor. An algebraic geometric analysis on the tensor manifold can lead to a characterization of the algebraic independent polynomials based on the sampling pattern which is related to the problem of data completion. Having understood the deterministic sampling conditions, the probabilistic conditions will be guaranteed to determine when the proposed deterministic conditions on the sampling patterns hold with high probability. The number of measurements needed to recover tensors are shown to be of a much lower order as compared to that when the data is converted to a matrix.