  Mean field glassy systems have a complicated energy landscape and an enormous
number of different Gibbs states. In this paper, we introduce a generalization
of the cavity method in order to describe the adiabatic evolution of these
glassy Gibbs states as an external parameter, such as the temperature, is
tuned. We give a general derivation of the method and describe in details the
solution of the resulting equations for the fully connected p-spin model, the
XOR-SAT problem and the anti-ferromagnetic Potts glass (or "coloring" problem).
As direct results of the states following method, we present a study of very
slow Monte-Carlo annealings, the demonstration of the presence of temperature
chaos in these systems, and the identification of a easy/hard transition for
simulated annealing in constraint optimization problems. We also discuss the
relation between our approach and the Franz-Parisi potential, as well as with
the reconstruction problem on trees in computer science. A mapping between the
states following method and the physics on the Nishimori line is also
presented.
