A random variable dominates another random variable with respect to the covariance order if the covariance of any two monotone increas- ing functions of this variable is smaller. We characterize completely the covariance order, give strong sufficient conditions for it, present a num- ber of examples in concrete economic applications, and provide natural extensions for the multivariate context. In analogy to mean preserving spreads in standard stochastic dominance, we show that the covariance order is intimately linked to a comparison of median preserving spreads of random variables. Moreover, it arises naturally in a variety of im- portant economic questions like, e.g., Hansen-Jagannathan stochastic discount factor bounds, the efficient portfolios implied by semi-variance optimization problems, or the measurement of macroeconomic inequal- ity and dispersion in beliefs.