Random partition models are a fundamental tool for Bayesian clustering and mixture modeling. Recent work has begun to treat the partition itself as a dynamic object, opening up new possibilities for modeling time-varying dependence structures in complex data, while also raising distinctive modeling, computational, and inferential challenges. In this talk, I will illustrate these opportunities and challenges through a few recent applications. Examples include local level dynamic random partition models for change point detection, which include a Markovian evolution of partitions within a state-space framework and couple it with non-marginal false discovery rate control; Bayesian temporal biclustering methods for multi-subject neuroscience studies, which jointly partition subjects and time into evolving profiles; and Bayesian semiparametric models that decode neuronal ensembles as spatially structured partitions of large populations of neurons from calcium imaging data. Together, these examples highlight both the promise of dynamic random partition models for representing evolving clustering structures and the open challenges on prior specification and scalable computation.