Cosmology with clusters of galaxies
  In this Chapter I review the role that galaxy clusters play as tools to constrain cosmological parameters. I will concentrate mostly on the application of the mass function of galaxy clusters, while other methods, such as that based on the baryon fraction, are covered by other Chapters of the book. Since most of the cosmological applications of galaxy clusters rely on precise measurements of their masses, a substantial part of my Lectures concentrates on the different methods that have been applied so far to weight galaxy clusters. I provide in Section 2 a short introduction to the basics of cosmic structure formation. In Section 3 I describe the Press--Schechter (PS) formalism to derive the cosmological mass function, then discussing extensions of the PS approach and the most recent calibrations from N--body simulations. In Section 4 I review the methods to build samples of galaxy clusters at different wavelengths. Section 5 is devoted to the discussion of different methods to derive cluster masses. In Section 6 I describe the cosmological constraints, which have been obtained so far by tracing the cluster mass function with a variety of methods. Finally, I describe in Section 7 the future perspectives for cosmology with galaxy clusters and the challenges for clusters to keep playing an important role in the era of precision cosmology. 
Stefano Borgani (Dept. of Astronomy, Univ. Trieste)