Since the product of two finitistic spaces need not be finitistic, and also because a continuous closed image of a finitistic space need not be finitistic, it is natural to enquire whether or not the class of finitistic spaces in closed under the formation of cones, reduced cones, suspensions, reduced suspensions, adjunction spaces, mapping cylinders, mapping cones, joins and smash products. In this paper we prove that all of the above constructs, except joins and smash products, of finitistic spaces are finitistic. The joins and smash products of finitistic spaces, however, need not be finitistic. We find sufficient conditions under which these are also finitistic.