The focus of this talk is 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 number of entries of higher-order moments of multivariate random variables scale exponentially with the order of the moments. For Gaussian Mixture Models (GMMs), we develop numerical methods forimplicit 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 $C^P$ tensor decomposition and sketch a recent algorithm which is faster than the state-of-the-art and comes with guarantees.