Authors’ abstract: Characterizations of subnormality due to Trent are extended to finite systems of operators, thus generalizing results of Găvruţă and Suciu (subnormal pairs) and Bînzar and Păunescu (subnormal triplets). Vector and operator multidimensional moment problems with jointly subnormal solutions are characterized entirely in terms of the initial data. Bounded as well as unbounded solutions are taken into consideration.