Finite-Set Statistics (FISST) is a direct generalization of conventional single-sensor, single-target statistics to the multisensor-multitarget realm. In particular, it deals with multitarget problems via multitarget Bayesian recursive nonlinear filtering (a direct generalization of the Bayesian recursive nonlinear filtering equations to the multitarget realm). The purpose of this paper is to (1) offer a brief bibliographical history of multitarget Bayesian recursive nonlinear filtering, and (2) describe the application of FISST techniques to the modeling of dynamic multitarget scenarios, e.g. scenarios in which targets can change mode or appear/disappear from one time-step to the next. Such problems can be addressed by FISST multitarget Markov motion models that take account of (among other things) the fact that the actual number of targets in a scenario (and not just the estimated number of targets) is a stochastic quantity--i.e., can randomly vary over time. We show that, in particular, there is a broad family of realistic multitarget density functions with the following property: If both the current multitarget posterior density and the multitarget Markov transition density belong to this family, then so does the time-update of the multitarget posterior. One result is potentially great computational savings in more general multitarget filtering problems. To better clarify some of conceptual key points underlying FISST, we also contrast it with an ad hoc approach, `generalized EAMLE,' and comment on `joint multitarget probabilities', a special case of certain core FISST concepts under a new name.