Stochastic dominance (SD) is a useful concept, especially in a multivariate context, where assessing multiattribute utility is challenging and different stakeholders might have divergent views. However, applying multivariate SD is difficult for three reasons: First, often distributions, even if fully known, cannot be ranked (e.g., by first-order SD). Second, easily verifiable integral conditions for multivariate SD often do not exist. Third, full information about multivariate distributions (including dependence structure) is difficult (and often impossible) to obtain. We will discuss how the concept of almost SD helps to overcome these challenges.