Several recent finance articles use the Omega measure (Keating and Shadwick, 2002), defined as a ratio of potential gains out of possible losses, for gauging the performance of funds or active strategies, in substitution of the traditional Sharpe ratio, with the arguments that return distributions are not Gaussian and volatility is not always the relevant risk metric. Other authors also use Omega for optimizing (non-linear) portfolios with important downside risk. However, we question in this article the relevance of such approaches. First, we show through a basic illustration that the Omega ratio is inconsistent with the Second-order Stochastic Dominance criterion. Furthermore, we observe that the trade-off between return and risk corresponding to the Omega measure, may be essentially influenced by the mean return. Next, we illustrate in static and dynamic frameworks that Omega-based optimal portfolios can be closely associated with classical optimization paradigms depending on the chosen threshold used in Omega. Finally, we present robustness checks on long-only asset and hedge fund databases, that confirm our results.