Abstract: In this talk, I will present some recent advances in the analytical study of the Keller-Segel(-Navier-Stokes) system, with a particular focus on the global existence and finite-time blowup of solutions. I will then discuss our recent work on the Keller-Segel-Navier-Stokes system with subquadratic logistic degradation in a three-dimensional smoothly bounded domain. This analysis is based on reasonably mild initial conditions and no-flux/no-flux/Dirichlet boundary conditions for the cell population, chemical, and fluid, respectively. Our findings indicate that persistent Dirac-type singularities can be ruled out by the logistic damping. Additionally, the eventual smoothness of these solutions will be demonstrated under the stricter condition that the linear growth coefficient of the population is not too large. This is joint work with Dr. Yu Tian at the Hong Kong Polytechnic University.