Chemostat refers to a laboratory device used for growing microorganisms in a cultured environment and has been regarded as an idealization of nature to study competition modelling in mathematical biology. The simplest form of chemostat model assumes that the availability of nutrient and its supply rate are both fixed. In addition, the tendency of microorganisms to adhere to surfaces is neglected by assuming the flow rate is fast enough. However, the previous assumptions largely limit the applicability of chemostat models to realistic competition systems. Because of this fact, biologists are really interested in stochastic chemostat models, deterministic models which are perturbed by some stochastic process and provide us a very good and realistic approximation of the real ones. Some stochastic chemostat models influenced by a brownian motion will be considered and analyzed by using the modern techniques concerning the theory of random dynamical systems (RDSs): interesting results related to existence and uniqueness of global solution just like global random attractors will be proved and several numerical simulations that support this work will be also shown.