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In this talk, I focus on the multivariate method of moments for parameter estimation. First from a theoretical standpoint, we show that in problems where the noise is high, the number of observations necessary to estimate parameters is dictated by the moments of the distribution. Second from a computational standpoint, we address the curse of dimensionality: the d-th moment of an n-dimensional random variable is a tensor with n^d entries. For Gaussian Mixture Models (GMMs), we develop numerical methods for implicit computations with the empirical moment tensors. This reduces the computational and storage costs, and opens the door to the competitiveness of the method of moments as compared to expectation maximization methods. Time permitting, we connect these results to symmetric CP tensor decomposition and sketch a recent algorithm which is faster than the state-of-the-art and comes with guarantees.