We estimate the global minimum variance (GMV) portfolio in the high-dimensional case using results from random matrix theory. This approach leads to a shrinkage-type estimator which is distribution-free and optimal in the sense of minimizing the out-of-sample variance. Its asymptotic properties are investigated assuming that the number of assets p depends on the sample size n such that p/n -> c is an element of (0, + infinity) as n tends to infinity. The results are obtained under weak assumptions imposed on the distribution of the asset returns: only the existence of the fourth moments is required. Furthermore, we make no assumption on the upper bound of the spectrum of the covariance matrix. As a result, the theoretical findings are also valid if the dependencies between the asset returns are described by a factor model which appears to be very popular in the financial literature nowadays. This is also documented in a numerical study where the small- and large-sample behavior of the derived estimator is compared with existing estimators of the GMV portfolio. The resulting estimator shows significant improvements and it turns out to be robust if the assumption of normality is violated.