The mathematical model of the chemostat has been extensively studied and extended from the eightees, not only as a mathematical representation of the chemostat device invented in the fifties, but also as a general model of resource/consumer dynamics in microbial ecosystems, such as in marine ecology, food fermentation, waste-water treatment, biotechnology. I will present a survey of some recent and less recent results about extensions of this model, that concern the roles of spatialization, density dependent growth, attachment/detachment and their impacts on stability and biodiversity.