In this talk we consider the limiting behavior of Tyler's and Maronna's M-estimators, in the regime that the number of samples n and the dimension p both go to infinity, and p/n converges to a constant y with 0<y<1. We prove that when the data samples are identically and independently generated from the Gaussian distribution N(0,I), the difference between the sample covariance matrix and a scaled version of Tyler's M-estimator or Maronna's M-estimator tends to zero in spectral norm, and the empirical spectral densities of both estimators converge to the Marcenko-Pastur distribution. We also prove that when the data samples are generated from an elliptical distribution, the limiting distribution of Tyler's M-estimator converges to a Marcenko-Pastur-Type distribution. This is joint work with Xiuyuan Cheng and Amit Singer.