  This article is concerned with classes of relational structures that are
closed under taking substructures and isomorphism, that have the joint
embedding property, and that furthermore have the Ramsey property, a strong
combinatorial property which resembles the statement of Ramsey's classic
theorem. Such classes of structures have been called Ramsey classes. Nesetril
and Roedl showed that they have the amalgamation property, and therefore each
such class has a homogeneous Fraisse-limit. Ramsey classes have recently
attracted attention due to a surprising link with the notion of extreme
amenability from topological dynamics. Other applications of Ramsey classes
include reduct classification of homogeneous structures.
  We give a survey of the various fundamental Ramsey classes and their (often
tricky) combinatorial proofs, and about various methods to derive new Ramsey
classes from known Ramsey classes. Finally, we state open problems related to a
potential classification of Ramsey classes.
