In this chapter, we present some examinations about Birnbaum importance measures of a group of components in a multi-state system with partially ordered state spaces, following previous works about importance measures for multi-state systems. We also present stochastic bounds for the multi-state Birnbaum importance measure when the stochastic performance of components is given by an associated probability measure. For binary-state reliability systems, various notions of importance measures are proposed, among which the Birnbaum importance measure is fundamental and usually defined as a partial differentiation of the reliability function under the assumption of stochastic independence among the components. However, the Birnbaum importance measure is equivalently defined to be the probability of the set of all the critical state vectors, where the independence assumption is not required. Such a way of thinking is easily extended to the multi-state case. In this chapter, we also discuss the ideas of stochastic dynamic importance measures of a group of components.