An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite-dimensional GIT, and conditions on the closure of orbits of almost-complex structures under the diffeomorphism group. Related analytic methods are also discussed, including estimates for energy functionals, Tian-Yau-Zelditch approximations, estimates for moment maps, complex Monge-Ampere equations and pluripotential theory, and the Kaehler-Ricci flow

85 pages, minor corrections added