  This article highlights historical achievements in the partition theory of
countable homogeneous relational structures, and presents recent work, current
trends, and open problems. Exciting recent developments include new methods
involving logic, topological Ramsey spaces, and category theory. The paper
concentrates on big Ramsey degrees, presenting their essential structure where
known and outlining areas for further development. Cognate areas, including
infinite dimensional Ramsey theory of homogeneous structures and partition
theory of uncountable structures, are also discussed.
