In this article we deal with three types of multivariate first-order stochastic dominance which serve for comparing random vectors. The first one is the strongest and it is generated by all non-decreasing multivariate utility functions. The second one, called weak multivariate stochastic dominance is defined by comparison of cumulative distribution functions of considered random vectors. The last one applies the univariate stochastic dominance notion to linear combinations of marginals. We compare these multivariate stochastic dominance relations among each other. Then we present important properties of strong and weak multivariate first-order stochastic dominance, in particular we describe their generators in the sense of von Neumann-Morgenstern utility functions and we explain their relation to joint and marginal distribution functions. Moreover, we discuss formulations of strong multivariate first-order stochastic dominance between random vectors with discrete distributions. Finally, we apply this stochastic dominance relation as a constraint to multivariate and multiperiod portfolio optimization problems.