This talk is related to several models of partial differential equations (PDEs) that describe the motion of microorganisms. In particular, I will focus on what is called "taxis" i.e. the motion of the microorganisms towards or away from an external stimulus. In general, "taxis" leads to aggregation whereas the diffusion have a tendency to spread the organisms. The competition between the diffusion and "taxis" will determine the formation of a singularity in finite time or a global solution for the associated PDE model.