We address the problem of estimating risk-minimizing portfolios from a sample of historical returns, when the underlying distribution that generates returns exhibits departures from the standard Gaussian assumption. Specifically, we examine how the underlying estimation problem is influenced by marginal heavy tails, as modeled by the univariate Student-t distribution, and multivariate tail-dependence, as modeled by the copula of a multivariate Student-t distribution. We show that when such departures from normality are present, robust alternatives to the classical variance portfolio estimator have lower risk.